Daily Papers

Training Large Language Models (LLMs) is memory-intensive due to the large number of parameters and associated optimization states. GaLore, a recent method, reduces memory usage by projecting weight gradients into a low-rank subspace without compromising performance. However, GaLore relies on time-consuming Singular Value Decomposition (SVD) operations to identify the subspace, and the frequent subspace updates lead to significant training time overhead. Moreover, GaLore offers minimal improvements in accuracy and efficiency compared to LoRA in more accessible fine-tuning scenarios. To address these limitations, we introduce Q-Galore, a novel approach that substantially reduces memory usage by combining quantization and low-rank projection, surpassing the benefits of GaLore. Our method is based on two key observations: (i) the gradient subspace exhibits diverse properties, with some layers converging early in training while others are subject to frequent changes; (ii) the projection matrices are highly resilient to low-bit quantization. Leveraging these insights, Q-GaLore adaptively updates the gradient subspace based on its convergence statistics, achieving comparable performance while significantly reducing the number of SVD operations. We maintain the projection matrices in INT4 format and weights in INT8 format, incorporating stochastic rounding to capture accumulated gradient information. This approach enables a high-precision training trajectory using only low-precision weights. We demonstrate that Q-GaLore achieves highly competitive performance with exceptional memory efficiency. At pre-training, Q-GaLore facilitates training a LLaMA-7B model from scratch on a single NVIDIA RTX 4060 Ti with only 16 GB memory. At fine-tuning, it reduces memory consumption by up to 50% compared to LoRA and GaLore, while consistently outperforming QLoRA at the same memory cost.

Large language models (LLMs) have revolutionized natural language understanding and generation but face significant memory bottlenecks during training. GaLore, Gradient Low-Rank Projection, addresses this issue by leveraging the inherent low-rank structure of weight gradients, enabling substantial memory savings without sacrificing performance. Recent works further extend GaLore from various aspects, including low-bit quantization and higher-order tensor structures. However, there are several remaining challenges for GaLore, such as the computational overhead of SVD for subspace updates and the integration with state-of-the-art training parallelization strategies (e.g., FSDP). In this paper, we present GaLore 2, an efficient and scalable GaLore framework that addresses these challenges and incorporates recent advancements. In addition, we demonstrate the scalability of GaLore 2 by pre-training Llama 7B from scratch using up to 500 billion training tokens, highlighting its potential impact on real LLM pre-training scenarios.

Recently, a wide range of memory-efficient LLM training algorithms have gained substantial popularity. These methods leverage the low-rank structure of gradients to project optimizer states into a subspace using projection matrix found by singular value decomposition (SVD). However, convergence of these algorithms is highly dependent on the update rules of their projection matrix. In this work, we provide the first convergence guarantee for arbitrary update rules of projection matrix. This guarantee is generally applicable to optimizers that can be analyzed with Hamiltonian Descent, including most common ones, such as LION, Adam. Inspired by our theoretical understanding, we propose Online Subspace Descent, a new family of subspace descent optimizer without SVD. Instead of updating the projection matrix with eigenvectors, Online Subspace Descent updates the projection matrix with online PCA. Online Subspace Descent is flexible and introduces only minimum overhead to training. We show that for the task of pretraining LLaMA models ranging from 60M to 7B parameters on the C4 dataset, Online Subspace Descent achieves lower perplexity and better downstream tasks performance than state-of-the-art low-rank training methods across different settings and narrows the gap with full-rank baselines.

Class-Incremental Learning (CIL) requires a learning system to continually learn new classes without forgetting. Despite the strong performance of Pre-Trained Models (PTMs) in CIL, a critical issue persists: learning new classes often results in the overwriting of old ones. Excessive modification of the network causes forgetting, while minimal adjustments lead to an inadequate fit for new classes. As a result, it is desired to figure out a way of efficient model updating without harming former knowledge. In this paper, we propose ExpAndable Subspace Ensemble (EASE) for PTM-based CIL. To enable model updating without conflict, we train a distinct lightweight adapter module for each new task, aiming to create task-specific subspaces. These adapters span a high-dimensional feature space, enabling joint decision-making across multiple subspaces. As data evolves, the expanding subspaces render the old class classifiers incompatible with new-stage spaces. Correspondingly, we design a semantic-guided prototype complement strategy that synthesizes old classes' new features without using any old class instance. Extensive experiments on seven benchmark datasets verify EASE's state-of-the-art performance. Code is available at: https://github.com/sun-hailong/CVPR24-Ease

Continual learning aims to learn multiple tasks sequentially while preserving prior knowledge, but faces the challenge of catastrophic forgetting when adapting to new tasks. Recently, approaches leveraging pre-trained models have gained increasing popularity in mitigating this issue, due to the strong generalization ability of foundation models. To adjust pre-trained models for new tasks, existing methods usually employ low-rank adaptation, which restricts parameter updates to a fixed low-rank subspace. However, constraining the optimization space inherently compromises the model's learning capacity, resulting in inferior performance. To address this limitation, we propose Continuous Subspace Optimization for Continual Learning (CoSO) to fine-tune the model in a series of subspaces rather than a single one. These sequential subspaces are dynamically determined through the singular value decomposition of the gradients. CoSO updates the model by projecting gradients onto these subspaces, ensuring memory-efficient optimization. To mitigate forgetting, the optimization subspace of each task is constrained to be orthogonal to the historical task subspace. During task learning, CoSO maintains a task-specific component that captures the critical update directions for the current task. Upon completing a task, this component is used to update the historical task subspace, laying the groundwork for subsequent learning. Extensive experiments on multiple datasets demonstrate that CoSO significantly outperforms state-of-the-art methods, especially in challenging scenarios with long task sequences.

We propose SPARC, a lightweight continual learning framework for large language models (LLMs) that enables efficient task adaptation through prompt tuning in a lower-dimensional space. By leveraging principal component analysis (PCA), we identify a compact subspace of the training data. Optimizing prompts in this lower-dimensional space enhances training efficiency, as it focuses updates on the most relevant features while reducing computational overhead. Furthermore, since the model's internal structure remains unaltered, the extensive knowledge gained from pretraining is fully preserved, ensuring that previously learned information is not compromised during adaptation. Our method achieves high knowledge retention in both task-incremental and domain-incremental continual learning setups while fine-tuning only 0.04% of the model's parameters. Additionally, by integrating LoRA, we enhance adaptability to computational constraints, allowing for a tradeoff between accuracy and training cost. Experiments on the SuperGLUE benchmark demonstrate that our PCA-based prompt tuning combined with LoRA maintains full knowledge retention while improving accuracy, utilizing only 1% of the model's parameters. These results establish our approach as a scalable and resource-efficient solution for continual learning in LLMs.

Low-rank optimization has emerged as a promising approach to enabling memory-efficient training of large language models (LLMs). Existing low-rank optimization methods typically project gradients onto a low-rank subspace, reducing the memory cost of storing optimizer states. A key challenge in these methods is selecting suitable subspaces to ensure an effective optimization trajectory. Most existing approaches select the dominant subspace to preserve gradient information, as this intuitively provides the best approximation. However, we find that in practice, the dominant subspace stops changing during pretraining, thereby constraining weight updates to similar subspaces. In this paper, we propose importance sampling for low-rank optimization in LLM pretraining with a provable convergence guarantee, which the dominant subspace approach does not have. Empirically, we demonstrate that our method significantly outperforms previous methods in LLM pretraining tasks.

Updating a truncated Singular Value Decomposition (SVD) is crucial in representation learning, especially when dealing with large-scale data matrices that continuously evolve in practical scenarios. Aligning SVD-based models with fast-paced updates becomes increasingly important. Existing methods for updating truncated SVDs employ Rayleigh-Ritz projection procedures, where projection matrices are augmented based on original singular vectors. However, these methods suffer from inefficiency due to the densification of the update matrix and the application of the projection to all singular vectors. To address these limitations, we introduce a novel method for dynamically approximating the truncated SVD of a sparse and temporally evolving matrix. Our approach leverages sparsity in the orthogonalization process of augmented matrices and utilizes an extended decomposition to independently store projections in the column space of singular vectors. Numerical experiments demonstrate a remarkable efficiency improvement of an order of magnitude compared to previous methods. Remarkably, this improvement is achieved while maintaining a comparable precision to existing approaches.

Policy gradient methods hold great potential for solving complex continuous control tasks. Still, their training efficiency can be improved by exploiting structure within the optimization problem. Recent work indicates that supervised learning can be accelerated by leveraging the fact that gradients lie in a low-dimensional and slowly-changing subspace. In this paper, we conduct a thorough evaluation of this phenomenon for two popular deep policy gradient methods on various simulated benchmark tasks. Our results demonstrate the existence of such gradient subspaces despite the continuously changing data distribution inherent to reinforcement learning. These findings reveal promising directions for future work on more efficient reinforcement learning, e.g., through improving parameter-space exploration or enabling second-order optimization.

We generalize the class vectors found in neural networks to linear subspaces (i.e.~points in the Grassmann manifold) and show that the Grassmann Class Representation (GCR) enables the simultaneous improvement in accuracy and feature transferability. In GCR, each class is a subspace and the logit is defined as the norm of the projection of a feature onto the class subspace. We integrate Riemannian SGD into deep learning frameworks such that class subspaces in a Grassmannian are jointly optimized with the rest model parameters. Compared to the vector form, the representative capability of subspaces is more powerful. We show that on ImageNet-1K, the top-1 error of ResNet50-D, ResNeXt50, Swin-T and Deit3-S are reduced by 5.6%, 4.5%, 3.0% and 3.5%, respectively. Subspaces also provide freedom for features to vary and we observed that the intra-class feature variability grows when the subspace dimension increases. Consequently, we found the quality of GCR features is better for downstream tasks. For ResNet50-D, the average linear transfer accuracy across 6 datasets improves from 77.98% to 79.70% compared to the strong baseline of vanilla softmax. For Swin-T, it improves from 81.5% to 83.4% and for Deit3, it improves from 73.8% to 81.4%. With these encouraging results, we believe that more applications could benefit from the Grassmann class representation. Code is released at https://github.com/innerlee/GCR.

In the training of large language models, parameter-efficient techniques such as LoRA optimize memory usage and reduce communication overhead and memory usage during the fine-tuning phase. However, applying such techniques directly during the pre-training phase results in poor performance, primarily because the premature implementation of low-rank training significantly reduces model accuracy. Existing methods like ReLoRA and GaLore have attempted to address this challenge by updating the low-rank subspace. However, they still fall short of achieving the accuracy of full-rank training. Specifically, ReLoRA restricts the frequency of updates to preserve optimizer states consistency, hindering its ability to closely approximate full-rank training behavior. Meanwhile, GaLore relies on Singular Value Decomposition (SVD) to approximate the full-rank space, which introduces accuracy loss during the approximation process. In this paper, we introduce SwitchLoRA, a parameter-efficient training technique that frequently and smoothly replaces the trainable parameters of LoRA adapters with alternative parameters. SwitchLoRA updates the low-rank subspace incrementally, targeting only a few dimensions at a time to minimize the impact on optimizer states. This allows a higher update frequency, thereby enhancing accuracy by enabling the updated parameters to more closely mimic full-rank behavior during the pre-training phase. Our results demonstrate that SwitchLoRA actually surpasses full-rank training, reducing perplexity from 15.23 to 15.01 on the LLaMA 1.3B model, while also cutting communication overhead by 54\% and memory usage by 13\%. Furthermore, after full fine-tuning the SwitchLoRA pre-trained model and the full-rank pre-trained model on the GLUE benchmark, the SwitchLoRA pre-trained model showed an average accuracy gain of about 1\% over the full-rank pre-trained model.

Large language models (LLMs) have demonstrated impressive generalization and emergent capabilities, yet their pre-training remains computationally expensive and sensitive to optimization dynamics. While Adam-based optimizers offer fast convergence by adapting learning rates coordinate-wise, recent studies reveal that their updates often suffer from poor spectral conditioning and low-rank structures, hindering efficiency. Muon addresses this issue via global spectral normalization but lacks the per-coordinate adaptivity of Adam. In this work, we propose Column-Normalized Adam (Conda), a novel optimizer that bridges the strengths of both approaches. Conda projects updates into an orthogonal subspace and applies column-wise second moment normalization based on the projected gradients, thereby achieving both improved spectral conditioning and maintaining coordinate-wise adaptivity. This design alleviates the spectral pathologies of Adam while preserving its fast convergence behavior. Extensive experiments on the LLaMA and GPT-2 series show that Conda consistently outperforms AdamW, Muon, and other baselines in pre-training. Remarkably, on the LLaMA series, Conda achieves 2-2.5 the convergence speed of AdamW, measured in both training steps and training time. Further ablations demonstrate its robustness under diverse training setups. These results collectively highlight Conda as an effective and broadly applicable optimizer for large-scale LLM training. The code is released on https://github.com/jie040109/Conda

Large pre-trained models are commonly adapted to downstream tasks using parameter-efficient fine-tuning methods such as Low-Rank Adaptation (LoRA), which injects small trainable low-rank matrices instead of updating all weights. While LoRA dramatically reduces trainable parameters with little overhead, it can still underperform full fine-tuning in accuracy and often converges more slowly. We introduce LoFT, a novel low-rank adaptation method that behaves like full fine-tuning by aligning the optimizer's internal dynamics with those of updating all model weights. LoFT not only learns weight updates in a low-rank subspace (like LoRA) but also properly projects the optimizer's first and second moments (Adam's momentum and variance) into the same subspace, mirroring full-model updates. By aligning the low-rank update itself with the full update, LoFT eliminates the need for tuning extra hyperparameters, e.g., LoRA scaling factor alpha. Empirically, this approach substantially narrows the performance gap between adapter-based tuning and full fine-tuning and consistently outperforms standard LoRA-style methods, all without increasing inference cost.

Federated Learning (FL) has gained significant attraction due to its ability to enable privacy-preserving training over decentralized data. Current literature in FL mostly focuses on single-task learning. However, over time, new tasks may appear in the clients and the global model should learn these tasks without forgetting previous tasks. This real-world scenario is known as Continual Federated Learning (CFL). The main challenge of CFL is Global Catastrophic Forgetting, which corresponds to the fact that when the global model is trained on new tasks, its performance on old tasks decreases. There have been a few recent works on CFL to propose methods that aim to address the global catastrophic forgetting problem. However, these works either have unrealistic assumptions on the availability of past data samples or violate the privacy principles of FL. We propose a novel method, Federated Orthogonal Training (FOT), to overcome these drawbacks and address the global catastrophic forgetting in CFL. Our algorithm extracts the global input subspace of each layer for old tasks and modifies the aggregated updates of new tasks such that they are orthogonal to the global principal subspace of old tasks for each layer. This decreases the interference between tasks, which is the main cause for forgetting. We empirically show that FOT outperforms state-of-the-art continual learning methods in the CFL setting, achieving an average accuracy gain of up to 15% with 27% lower forgetting while only incurring a minimal computation and communication cost.

LLM agents are widely deployed in complex interactive tasks, yet privacy constraints often preclude centralized optimization and co-evolution across dynamic environments. While Federated Learning (FL) has proven effective on static datasets, its extension to the open-ended self-evolution of agents remains underexplored. Directly applying standard FL is challenging: heterogeneous tasks and sparse, trajectory-level rewards introduce severe gradient conflicts, destabilizing the global optimization process. To bridge this gap, we propose Fed-SE, a Federated Self-Evolution framework for LLM agents. Fed-SE establishes a local evolution-global aggregation paradigm. Locally, agents employ parameter-efficient fine-tuning on filtered, high-return trajectories to achieve stable gradient updates. Globally, Fed-SE aggregates updates within a low-rank subspace that disentangles environment-specific dynamics, effectively reducing negative transfer across clients. Experiments across five heterogeneous environments demonstrate that Fed-SE improves average task success rates by approximately 18% over federated baselines, validating its effectiveness in robust cross-environment knowledge transfer in privacy-constrained deployments.

Recent advances in reasoning capabilities of large language models (LLMs) are largely driven by reinforcement learning (RL), yet the underlying parameter dynamics during RL training remain poorly understood. This work identifies two fundamental properties of RL-induced parameter updates in LLMs: (1) Rank-1 Dominance, where the top singular subspace of the parameter update matrix nearly fully determines reasoning improvements, recovering over 99\% of performance gains; and (2) Rank-1 Linear Dynamics, where this dominant subspace evolves linearly throughout training, enabling accurate prediction from early checkpoints. Extensive experiments across 8 LLMs and 7 algorithms validate the generalizability of these properties. More importantly, based on these findings, we propose AlphaRL, a plug-in acceleration framework that extrapolates the final parameter update using a short early training window, achieving up to 2.5 speedup while retaining \textgreater 96\% of reasoning performance without extra modules or hyperparameter tuning. This positions our finding as a versatile and practical tool for large-scale RL, opening a path toward principled, interpretable, and efficient training paradigm for LLMs.

Fine-tuning large language models (LMs) for individual tasks yields strong performance but is expensive for deployment and storage. Recent works explore model merging to combine multiple task-specific models into a single multi-task model without additional training. However, existing merging methods often fail for models fine-tuned with low-rank adaptation (LoRA), due to significant performance degradation. In this paper, we show that this issue arises from a previously overlooked interplay between model parameters and data distributions. We propose Orthogonal Subspaces for Robust model Merging (OSRM) to constrain the LoRA subspace *prior* to fine-tuning, ensuring that updates relevant to one task do not adversely shift outputs for others. Our approach can seamlessly integrate with most existing merging algorithms, reducing the unintended interference among tasks. Extensive experiments on eight datasets, tested with three widely used LMs and two large LMs, demonstrate that our method not only boosts merging performance but also preserves single-task accuracy. Furthermore, our approach exhibits greater robustness to the hyperparameters of merging. These results highlight the importance of data-parameter interaction in model merging and offer a plug-and-play solution for merging LoRA models.

A vast amount of instruction tuning data is crucial for the impressive performance of Large Multimodal Models (LMMs), but the associated computational costs and data collection demands during supervised fine-tuning make it impractical for most researchers. Federated learning (FL) has the potential to leverage all distributed data and training resources to reduce the overhead of joint training. However, most existing methods assume a fixed number of tasks, while in real-world scenarios, clients continuously encounter new knowledge and often struggle to retain old tasks due to memory constraints. In this work, we introduce the Federated Continual Instruction Tuning (FCIT) benchmark to model this real-world challenge. Our benchmark includes two realistic scenarios, encompassing four different settings and twelve carefully curated instruction tuning datasets. To address the challenges posed by FCIT, we propose dynamic knowledge organization to effectively integrate updates from different tasks during training and subspace selective activation to allocate task-specific output during inference. Extensive experimental results demonstrate that our proposed method significantly enhances model performance across varying levels of data heterogeneity and catastrophic forgetting. Code and dataset are released at https://github.com/Ghy0501/FCIT.

Training large language models (LLMs) is often bottlenecked by extreme memory demands, with optimizer states dominating the footprint. Recent works mitigates this cost by projecting gradients into low-dimensional subspaces using sophisticated update strategies. In this paper, we analyze the dynamics of gradient space and its underlying subspaces. We find that while a small subspace captures most gradient energy, a significant portion still resides in the residual bulk; moreover, the influence of the core subspace diminishes over time and in deeper layers. We also observe that the gradient space exhibits near-flat curvature, calling for algorithms that explicitly account for this geometry. Motivated by these insights, we introduce a suite of randomized algorithms, GrassWalk and GrassJump, which exploit subspace and achieve state-of-the-art memory savings while improving performance on LLaMA-1B and LLaMA-7B pretraining.

Adversarial robustness is essential for security and reliability of machine learning systems. However, adversarial robustness enhanced by defense algorithms is easily erased as the neural network's weights update to learn new tasks. To address this vulnerability, it is essential to improve the capability of neural networks in terms of robust continual learning. Specially, we propose a novel gradient projection technique that effectively stabilizes sample gradients from previous data by orthogonally projecting back-propagation gradients onto a crucial subspace before using them for weight updates. This technique can maintaining robustness by collaborating with a class of defense algorithms through sample gradient smoothing. The experimental results on four benchmarks including Split-CIFAR100 and Split-miniImageNet, demonstrate that the superiority of the proposed approach in mitigating rapidly degradation of robustness during continual learning even when facing strong adversarial attacks.

We tackle the problem disentangling the latent space of an autoencoder in order to separate labelled attribute information from other characteristic information. This then allows us to change selected attributes while preserving other information. Our method, matrix subspace projection, is much simpler than previous approaches to latent space factorisation, for example not requiring multiple discriminators or a careful weighting among their loss functions. Furthermore our new model can be applied to autoencoders as a plugin, and works across diverse domains such as images or text. We demonstrate the utility of our method for attribute manipulation in autoencoders trained across varied domains, using both human evaluation and automated methods. The quality of generation of our new model (e.g. reconstruction, conditional generation) is highly competitive to a number of strong baselines.

Projection maintenance is one of the core data structure tasks. Efficient data structures for projection maintenance have led to recent breakthroughs in many convex programming algorithms. In this work, we further extend this framework to the Kronecker product structure. Given a constraint matrix {sf A} and a positive semi-definite matrix Win R^{ntimes n} with a sparse eigenbasis, we consider the task of maintaining the projection in the form of {sf B}^top({sf B}{sf B}^top)^{-1}{sf B}, where {sf B}={sf A}(Wotimes I) or {sf B}={sf A}(W^{1/2}otimes W^{1/2}). At each iteration, the weight matrix W receives a low rank change and we receive a new vector h. The goal is to maintain the projection matrix and answer the query {sf B}^top({sf B}{sf B}^top)^{-1}{sf B}h with good approximation guarantees. We design a fast dynamic data structure for this task and it is robust against an adaptive adversary. Following the beautiful and pioneering work of [Beimel, Kaplan, Mansour, Nissim, Saranurak and Stemmer, STOC'22], we use tools from differential privacy to reduce the randomness required by the data structure and further improve the running time.

Residual connections are pivotal for deep neural networks, enabling greater depth by mitigating vanishing gradients. However, in standard residual updates, the module's output is directly added to the input stream. This can lead to updates that predominantly reinforce or modulate the existing stream direction, potentially underutilizing the module's capacity for learning entirely novel features. In this work, we introduce Orthogonal Residual Update: we decompose the module's output relative to the input stream and add only the component orthogonal to this stream. This design aims to guide modules to contribute primarily new representational directions, fostering richer feature learning while promoting more efficient training. We demonstrate that our orthogonal update strategy improves generalization accuracy and training stability across diverse architectures (ResNetV2, Vision Transformers) and datasets (CIFARs, TinyImageNet, ImageNet-1k), achieving, for instance, a +4.3\%p top-1 accuracy gain for ViT-B on ImageNet-1k.

Merging models fine-tuned from a common, extensively pre-trained large model but specialized for different tasks has been demonstrated as a cheap and scalable strategy to construct a multi-task model that performs well across diverse tasks. Recent research, exemplified by task arithmetic, highlights that this multi-task model can be derived through arithmetic operations on task vectors. Nevertheless, current merging techniques frequently resolve potential conflicts among parameters from task-specific models by evaluating individual attributes, such as the parameters' magnitude or sign, overlooking their collective impact on the overall functionality of the model. In this work, we propose the CONtinuous relaxation of disCRETE (Concrete) subspace learning method to identify a common low-dimensional subspace and utilize its shared information to track the interference problem without sacrificing much performance. Specifically, we model the problem as a bi-level optimization problem and introduce a meta-learning framework to find the Concrete subspace mask through gradient-based techniques. At the upper level, we focus on learning a shared Concrete mask to identify the subspace, while at the inner level, model merging is performed to maximize the performance of the merged model. We conduct extensive experiments on both vision domain and language domain, and the results demonstrate the effectiveness of our method. The code is available at https://github.com/tanganke/subspace_fusion

We consider the problem of simultaneously clustering and learning a linear representation of data lying close to a union of low-dimensional manifolds, a fundamental task in machine learning and computer vision. When the manifolds are assumed to be linear subspaces, this reduces to the classical problem of subspace clustering, which has been studied extensively over the past two decades. Unfortunately, many real-world datasets such as natural images can not be well approximated by linear subspaces. On the other hand, numerous works have attempted to learn an appropriate transformation of the data, such that data is mapped from a union of general non-linear manifolds to a union of linear subspaces (with points from the same manifold being mapped to the same subspace). However, many existing works have limitations such as assuming knowledge of the membership of samples to clusters, requiring high sampling density, or being shown theoretically to learn trivial representations. In this paper, we propose to optimize the Maximal Coding Rate Reduction metric with respect to both the data representation and a novel doubly stochastic cluster membership, inspired by state-of-the-art subspace clustering results. We give a parameterization of such a representation and membership, allowing efficient mini-batching and one-shot initialization. Experiments on CIFAR-10, -20, -100, and TinyImageNet-200 datasets show that the proposed method is much more accurate and scalable than state-of-the-art deep clustering methods, and further learns a latent linear representation of the data.

Reinforcement learning (RL) yields substantial improvements in large language models (LLMs) downstream task performance and alignment with human values. Surprisingly, such large gains result from updating only a small subnetwork comprising just 5 percent to 30 percent of the parameters, with the rest effectively unchanged. We refer to this phenomenon as parameter update sparsity induced by RL. It is observed across all 7 widely used RL algorithms (e.g., PPO, GRPO, DPO) and all 10 LLMs from different families in our experiments. This sparsity is intrinsic and occurs without any explicit sparsity promoting regularizations or architectural constraints. Finetuning the subnetwork alone recovers the test accuracy, and, remarkably, produces a model nearly identical to the one obtained via full finetuning. The subnetworks from different random seeds, training data, and even RL algorithms show substantially greater overlap than expected by chance. Our analysis suggests that this sparsity is not due to updating only a subset of layers, instead, nearly all parameter matrices receive similarly sparse updates. Moreover, the updates to almost all parameter matrices are nearly full-rank, suggesting RL updates a small subset of parameters that nevertheless span almost the full subspaces that the parameter matrices can represent. We conjecture that the this update sparsity can be primarily attributed to training on data that is near the policy distribution, techniques that encourage the policy to remain close to the pretrained model, such as the KL regularization and gradient clipping, have limited impact.

Orthogonal gradient updates have emerged as a promising direction in optimization for machine learning. However, traditional approaches such as SVD/QR decomposition incur prohibitive computational costs of O(n^3) and underperform compared to well-tuned SGD with momentum, since momentum is applied only after strict orthogonalization. Recent advances, such as Muon, improve efficiency by applying momentum before orthogonalization and producing semi-orthogonal matrices via Newton-Schulz iterations, reducing complexity to O(n^2). Nevertheless, quadratic costs remain a bottleneck. In this work, we study the semi-orthogonal properties of momentum-based updates and develop a method to bound momentum updates under a spectral-norm trust region, preserving directional information without requiring explicit semi-orthogonalization. We propose AuON (Alternative Unit-norm momentum updates by Normalized nonlinear scaling), a linear-time optimizer that achieves strong performance without constructing semi-orthogonal matrices, while preserving structural alignment and reconditioning ill-posed updates. Our approach combines hyperbolic-cosine RMS scaling transformations with normalization, demonstrating both effectiveness and computational efficiency compared to Newton-Schulz methods. We further introduce a hybrid variant (Hybrid-AuON) that applies a single Newton-Schulz iteration. Experiments across vision and language benchmarks show that AuON and its hybrid variant achieve performance comparable to strong baselines such as AdamW and Muon. Code is available at: https://github.com/ryyzn9/AuON

Mechanistic interpretability aims to understand model behaviors in terms of specific, interpretable features, often hypothesized to manifest as low-dimensional subspaces of activations. Specifically, recent studies have explored subspace interventions (such as activation patching) as a way to simultaneously manipulate model behavior and attribute the features behind it to given subspaces. In this work, we demonstrate that these two aims diverge, potentially leading to an illusory sense of interpretability. Counterintuitively, even if a subspace intervention makes the model's output behave as if the value of a feature was changed, this effect may be achieved by activating a dormant parallel pathway leveraging another subspace that is causally disconnected from model outputs. We demonstrate this phenomenon in a distilled mathematical example, in two real-world domains (the indirect object identification task and factual recall), and present evidence for its prevalence in practice. In the context of factual recall, we further show a link to rank-1 fact editing, providing a mechanistic explanation for previous work observing an inconsistency between fact editing performance and fact localization. However, this does not imply that activation patching of subspaces is intrinsically unfit for interpretability. To contextualize our findings, we also show what a success case looks like in a task (indirect object identification) where prior manual circuit analysis informs an understanding of the location of a feature. We explore the additional evidence needed to argue that a patched subspace is faithful.

Model merging integrates the weights of multiple task-specific models into a single multi-task model. Despite recent interest in the problem, a significant performance gap between the combined and single-task models remains. In this paper, we investigate the key characteristics of task matrices -- weight update matrices applied to a pre-trained model -- that enable effective merging. We show that alignment between singular components of task-specific and merged matrices strongly correlates with performance improvement over the pre-trained model. Based on this, we propose an isotropic merging framework that flattens the singular value spectrum of task matrices, enhances alignment, and reduces the performance gap. Additionally, we incorporate both common and task-specific subspaces to further improve alignment and performance. Our proposed approach achieves state-of-the-art performance across multiple scenarios, including various sets of tasks and model scales. This work advances the understanding of model merging dynamics, offering an effective methodology to merge models without requiring additional training. Code is available at https://github.com/danielm1405/iso-merging .

In several recently proposed stochastic optimization methods (e.g. RMSProp, Adam, Adadelta), parameter updates are scaled by the inverse square roots of exponential moving averages of squared past gradients. Maintaining these per-parameter second-moment estimators requires memory equal to the number of parameters. For the case of neural network weight matrices, we propose maintaining only the per-row and per-column sums of these moving averages, and estimating the per-parameter second moments based on these sums. We demonstrate empirically that this method produces similar results to the baseline. Secondly, we show that adaptive methods can produce larger-than-desired updates when the decay rate of the second moment accumulator is too slow. We propose update clipping and a gradually increasing decay rate scheme as remedies. Combining these methods and dropping momentum, we achieve comparable results to the published Adam regime in training the Transformer model on the WMT 2014 English-German machine translation task, while using very little auxiliary storage in the optimizer. Finally, we propose scaling the parameter updates based on the scale of the parameters themselves.

Traditional white-box methods for creating adversarial perturbations against LLMs typically rely only on gradient computation from the targeted model, ignoring the internal mechanisms responsible for attack success or failure. Conversely, interpretability studies that analyze these internal mechanisms lack practical applications beyond runtime interventions. We bridge this gap by introducing a novel white-box approach that leverages mechanistic interpretability techniques to craft practical adversarial inputs. Specifically, we first identify acceptance subspaces - sets of feature vectors that do not trigger the model's refusal mechanisms - then use gradient-based optimization to reroute embeddings from refusal subspaces to acceptance subspaces, effectively achieving jailbreaks. This targeted approach significantly reduces computation cost, achieving attack success rates of 80-95\% on state-of-the-art models including Gemma2, Llama3.2, and Qwen2.5 within minutes or even seconds, compared to existing techniques that often fail or require hours of computation. We believe this approach opens a new direction for both attack research and defense development. Furthermore, it showcases a practical application of mechanistic interpretability where other methods are less efficient, which highlights its utility. The code and generated datasets are available at https://github.com/Sckathach/subspace-rerouting.

Driven by the rapid growth of model parameters, parameter-efficient fine-tuning (PEFT) has become essential for adapting large models to diverse downstream tasks under constrained computational resources. Within this paradigm, orthogonal fine-tuning and its variants preserve semantic representations of pre-trained models, but struggle to achieve both expressiveness and efficiency in terms of parameter counts, memory, and computation. To overcome this limitation, we propose efficient Orthogonal Fine-Tuning with Principal Subspace adaptation (PSOFT), which confines orthogonal transformations to the principal subspace of pre-trained weights. Specifically, PSOFT constructs this subspace via matrix decomposition to enable compatible transformations with higher effective rank, establishes a theoretical condition that strictly maintains the geometry of this subspace for essential semantic preservation, and introduces efficient tunable vectors that gradually relax orthogonality during training to enhance adaptability. Extensive experiments on 35 NLP and CV tasks across four representative models demonstrate that PSOFT offers a practical and scalable solution to simultaneously achieve semantic preservation, expressiveness, and multi-dimensional efficiency in PEFT. The code is publicly available at https://github.com/fei407/PSOFT.

Adapting pre-trained foundation models for various downstream tasks has been prevalent in artificial intelligence. Due to the vast number of tasks and high costs, adjusting all parameters becomes unfeasible. To mitigate this, several fine-tuning techniques have been developed to update the pre-trained model weights in a more resource-efficient manner, such as through low-rank adjustments. Yet, almost all of these methods focus on linear weights, neglecting the intricacies of parameter spaces in higher dimensions like 4D. Alternatively, some methods can be adapted for high-dimensional parameter space by compressing changes in the original space into two dimensions and then employing low-rank matrix decomposition. However, these approaches destructs the structural integrity of the involved high-dimensional spaces. To tackle the diversity of dimensional spaces across different foundation models and provide a more precise representation of the changes within these spaces, this paper introduces a generalized parameter-efficient fine-tuning framework, FLoRA, designed for various dimensional parameter space. Specifically, utilizing Tucker decomposition, FLoRA asserts that changes in each dimensional parameter space are based on a low-rank core space which maintains the consistent topological structure with the original space. It then models the changes through this core space alongside corresponding weights to reconstruct alterations in the original space. FLoRA effectively preserves the structural integrity of the change of original N-dimensional parameter space, meanwhile decomposes it via low-rank tensor decomposition. Extensive experiments on computer vision, natural language processing and multi-modal tasks validate FLoRA's effectiveness. Codes are available at https://github.com/SJTU-DeepVisionLab/FLoRA.

Given a matrix Min R^{mtimes n}, the low rank matrix completion problem asks us to find a rank-k approximation of M as UV^top for Uin R^{mtimes k} and Vin R^{ntimes k} by only observing a few entries specified by a set of entries Omegasubseteq [m]times [n]. In particular, we examine an approach that is widely used in practice -- the alternating minimization framework. Jain, Netrapalli and Sanghavi~jns13 showed that if M has incoherent rows and columns, then alternating minimization provably recovers the matrix M by observing a nearly linear in n number of entries. While the sample complexity has been subsequently improved~glz17, alternating minimization steps are required to be computed exactly. This hinders the development of more efficient algorithms and fails to depict the practical implementation of alternating minimization, where the updates are usually performed approximately in favor of efficiency. In this paper, we take a major step towards a more efficient and error-robust alternating minimization framework. To this end, we develop an analytical framework for alternating minimization that can tolerate moderate amount of errors caused by approximate updates. Moreover, our algorithm runs in time widetilde O(|Omega| k), which is nearly linear in the time to verify the solution while preserving the sample complexity. This improves upon all prior known alternating minimization approaches which require widetilde O(|Omega| k^2) time.

We introduce the Subspace Power Method (SPM) for calculating the CP decomposition of low-rank real symmetric tensors. This algorithm calculates one new CP component at a time, alternating between applying the shifted symmetric higher-order power method (SS-HOPM) to a certain modified tensor, constructed from a matrix flattening of the original tensor; and using appropriate deflation steps. We obtain rigorous guarantees for SPM regarding convergence and global optima for input tensors of dimension d and order m of CP rank up to O(d^{lfloor m/2rfloor}), via results in classical algebraic geometry and optimization theory. As a by-product of our analysis we prove that SS-HOPM converges unconditionally, settling a conjecture in [Kolda, T.G., Mayo, J.R.: Shifted power method for computing tensor eigenpairs. SIAM Journal on Matrix Analysis and Applications 32(4), 1095-1124 (2011)]. We present numerical experiments which demonstrate that SPM is efficient and robust to noise, being up to one order of magnitude faster than state-of-the-art CP decomposition algorithms in certain experiments. Furthermore, prior knowledge of the CP rank is not required by SPM.

We show that in a variety of large-scale deep learning scenarios the gradient dynamically converges to a very small subspace after a short period of training. The subspace is spanned by a few top eigenvectors of the Hessian (equal to the number of classes in the dataset), and is mostly preserved over long periods of training. A simple argument then suggests that gradient descent may happen mostly in this subspace. We give an example of this effect in a solvable model of classification, and we comment on possible implications for optimization and learning.

Retrieval systems rely on representations learned by increasingly powerful models. However, due to the high training cost and inconsistencies in learned representations, there is significant interest in facilitating communication between representations and ensuring compatibility across independently trained neural networks. In the literature, two primary approaches are commonly used to adapt different learned representations: affine transformations, which adapt well to specific distributions but can significantly alter the original representation, and orthogonal transformations, which preserve the original structure with strict geometric constraints but limit adaptability. A key challenge is adapting the latent spaces of updated models to align with those of previous models on downstream distributions while preserving the newly learned representation spaces. In this paper, we impose a relaxed orthogonality constraint, namely λ-Orthogonality regularization, while learning an affine transformation, to obtain distribution-specific adaptation while retaining the original learned representations. Extensive experiments across various architectures and datasets validate our approach, demonstrating that it preserves the model's zero-shot performance and ensures compatibility across model updates. Code available at: https://github.com/miccunifi/lambda_orthogonality.git{https://github.com/miccunifi/lambda\_orthogonality}.

Model customization necessitates high-quality and diverse datasets, but acquiring such data remains time-consuming and labor-intensive. Despite the great potential of large language models (LLMs) for data synthesis, current approaches are constrained by limited seed data, model biases, and low-variation prompts, resulting in limited diversity and biased distributions with the increase of data scales. To tackle this challenge, we introduce TREESYNTH, a tree-guided subspace-based data synthesis approach inspired by decision trees. It constructs a spatial partitioning tree to recursively divide a task-specific full data space (i.e., root node) into numerous atomic subspaces (i.e., leaf nodes) with mutually exclusive and exhaustive attributes to ensure both distinctiveness and comprehensiveness before synthesizing samples within each atomic subspace. This globally dividing-and-synthesizing method finally collects subspace samples into a comprehensive dataset, effectively circumventing repetition and space collapse to ensure the diversity of large-scale data synthesis. Furthermore, the spatial partitioning tree enables sample allocation into atomic subspaces, allowing the rebalancing of existing datasets for more balanced and comprehensive distributions. Empirically, extensive experiments across diverse benchmarks consistently demonstrate the superior data diversity, model performance, and robust scalability of TREESYNTH compared to both human-crafted datasets and peer data synthesis methods, with an average performance gain reaching 10%. Besides, the consistent improvements of TREESYNTH-balanced datasets highlight its efficacious application to redistribute existing datasets for more comprehensive coverage and the induced performance enhancement. The code is available at https://github.com/cpa2001/TreeSynth.

With the increase in the number of parameters in large language models, the process of pre-training and fine-tuning increasingly demands larger volumes of GPU memory. A significant portion of this memory is typically consumed by the optimizer state. To overcome this challenge, recent approaches such as low-rank adaptation (LoRA (Hu et al., 2021)), low-rank gradient projection (GaLore (Zhao et al., 2024)), and blockwise optimization (BAdam (Luo et al., 2024)) have been proposed. However, in all these algorithms, the effective rank of the weight updates remains low-rank, which can lead to a substantial loss of information from the gradient. This loss can be critically important, especially during the pre-training stage. In this paper, we introduce FRUGAL (Full-Rank Updates with GrAdient spLitting), a new memory-efficient optimization framework. FRUGAL leverages gradient splitting to perform low-dimensional updates using advanced algorithms (such as Adam), while updates along the remaining directions are executed via state-free methods like SGD or signSGD (Bernstein et al., 2018). Our framework can be integrated with various low-rank update selection techniques, including GaLore and BAdam. We provide theoretical convergence guarantees for our framework when using SGDM for low-dimensional updates and SGD for state-free updates. Additionally, our method consistently outperforms concurrent approaches across various fixed memory budgets, achieving state-of-the-art results in pre-training and fine-tuning tasks while balancing memory efficiency and performance metrics.

We introduce two complementary techniques for efficient adaptive optimization that reduce memory requirements while accelerating training of large-scale neural networks. The first technique, Subset-Norm adaptive step size, generalizes AdaGrad-Norm and AdaGrad(-Coordinate) by reducing the second moment term's memory footprint from O(d) to O(d) through step-size sharing, where d is the model size. For non-convex smooth objectives under coordinate-wise sub-gaussian gradient noise, we prove a noise-adapted high-probability convergence guarantee showing improved dimensional dependence over existing methods. Our second technique, Subspace-Momentum, reduces the momentum state's memory footprint by operating in a low-dimensional subspace while applying standard SGD in the orthogonal complement. We establish high-probability convergence rates under similar relaxed assumptions. Empirical evaluation on LLaMA models from 60M to 1B parameters demonstrates the effectiveness of our methods, where combining subset-norm with subspace-momentum achieves Adam's validation perplexity in approximately half the training tokens (6.8B vs 13.1B) while using only 20% of the Adam's optimizer-states memory footprint and requiring minimal additional hyperparameter tuning.

Learning similarity is a key aspect in medical image analysis, particularly in recommendation systems or in uncovering the interpretation of anatomical data in images. Most existing methods learn such similarities in the embedding space over image sets using a single metric learner. Images, however, have a variety of object attributes such as color, shape, or artifacts. Encoding such attributes using a single metric learner is inadequate and may fail to generalize. Instead, multiple learners could focus on separate aspects of these attributes in subspaces of an overarching embedding. This, however, implies the number of learners to be found empirically for each new dataset. This work, Dynamic Subspace Learners, proposes to dynamically exploit multiple learners by removing the need of knowing apriori the number of learners and aggregating new subspace learners during training. Furthermore, the visual interpretability of such subspace learning is enforced by integrating an attention module into our method. This integrated attention mechanism provides a visual insight of discriminative image features that contribute to the clustering of image sets and a visual explanation of the embedding features. The benefits of our attention-based dynamic subspace learners are evaluated in the application of image clustering, image retrieval, and weakly supervised segmentation. Our method achieves competitive results with the performances of multiple learners baselines and significantly outperforms the classification network in terms of clustering and retrieval scores on three different public benchmark datasets. Moreover, our attention maps offer a proxy-labels, which improves the segmentation accuracy up to 15% in Dice scores when compared to state-of-the-art interpretation techniques.

Many recently trained neural networks employ large numbers of parameters to achieve good performance. One may intuitively use the number of parameters required as a rough gauge of the difficulty of a problem. But how accurate are such notions? How many parameters are really needed? In this paper we attempt to answer this question by training networks not in their native parameter space, but instead in a smaller, randomly oriented subspace. We slowly increase the dimension of this subspace, note at which dimension solutions first appear, and define this to be the intrinsic dimension of the objective landscape. The approach is simple to implement, computationally tractable, and produces several suggestive conclusions. Many problems have smaller intrinsic dimensions than one might suspect, and the intrinsic dimension for a given dataset varies little across a family of models with vastly different sizes. This latter result has the profound implication that once a parameter space is large enough to solve a problem, extra parameters serve directly to increase the dimensionality of the solution manifold. Intrinsic dimension allows some quantitative comparison of problem difficulty across supervised, reinforcement, and other types of learning where we conclude, for example, that solving the inverted pendulum problem is 100 times easier than classifying digits from MNIST, and playing Atari Pong from pixels is about as hard as classifying CIFAR-10. In addition to providing new cartography of the objective landscapes wandered by parameterized models, the method is a simple technique for constructively obtaining an upper bound on the minimum description length of a solution. A byproduct of this construction is a simple approach for compressing networks, in some cases by more than 100 times.

Continual learning in large language models (LLMs) is prone to catastrophic forgetting, where adapting to new tasks significantly degrades performance on previously learned ones. Existing methods typically rely on low-rank, parameter-efficient updates that limit the model's expressivity and introduce additional parameters per task, leading to scalability issues. To address these limitations, we propose a novel continual full fine-tuning approach leveraging adaptive singular value decomposition (SVD). Our method dynamically identifies task-specific low-rank parameter subspaces and constrains updates to be orthogonal to critical directions associated with prior tasks, thus effectively minimizing interference without additional parameter overhead or storing previous task gradients. We evaluate our approach extensively on standard continual learning benchmarks using both encoder-decoder (T5-Large) and decoder-only (LLaMA-2 7B) models, spanning diverse tasks including classification, generation, and reasoning. Empirically, our method achieves state-of-the-art results, up to 7% higher average accuracy than recent baselines like O-LoRA, and notably maintains the model's general linguistic capabilities, instruction-following accuracy, and safety throughout the continual learning process by reducing forgetting to near-negligible levels. Our adaptive SVD framework effectively balances model plasticity and knowledge retention, providing a practical, theoretically grounded, and computationally scalable solution for continual learning scenarios in large language models.

In this article, we develop an asymptotic method for constructing confidence regions for the set of all linear subspaces arising from PCA, from which we derive hypothesis tests on this set. Our method is based on the geometry of Riemannian manifolds with which some sets of linear subspaces are endowed.

Deep model merging represents an emerging research direction that combines multiple fine-tuned models to harness their specialized capabilities across different tasks and domains. Current model merging techniques focus on merging all available models simultaneously, with weight interpolation-based methods being the predominant approaches. However, these conventional approaches are not well-suited for scenarios where models become available sequentially, and they often suffer from high memory requirements and potential interference between tasks. In this study, we propose a training-free projection-based continual merging method that processes models sequentially through orthogonal projections of weight matrices and adaptive scaling mechanisms. Our method operates by projecting new parameter updates onto subspaces orthogonal to existing merged parameter updates while using an adaptive scaling mechanism to maintain stable parameter distances, enabling efficient sequential integration of task-specific knowledge. Our approach maintains constant memory complexity to the number of models, minimizes interference between tasks through orthogonal projections, and retains the performance of previously merged models through adaptive task vector scaling. Extensive experiments on CLIP-ViT models demonstrate that our method achieves a 5-8% average accuracy improvement while maintaining robust performance in different task orderings.

Low-rank adapters have become standard for efficiently fine-tuning large language models (LLMs), but they often fall short of achieving the performance of full fine-tuning. We propose a method, LoRA Silver Bullet or LoRA-SB, that approximates full fine-tuning within low-rank subspaces using a carefully designed initialization strategy. We theoretically demonstrate that the architecture of LoRA-XS, which inserts a learnable (r x r) matrix between B and A while keeping other matrices fixed, provides the precise conditions needed for this approximation. We leverage its constrained update space to achieve optimal scaling for high-rank gradient updates while removing the need for hyperparameter tuning. We prove that our initialization offers an optimal low-rank approximation of the initial gradient and preserves update directions throughout training. Extensive experiments across mathematical reasoning, commonsense reasoning, and language understanding tasks demonstrate that our approach exceeds the performance of standard LoRA while using 27-90 times fewer learnable parameters, and comprehensively outperforms LoRA-XS. Our findings establish that it is possible to simulate full fine-tuning in low-rank subspaces, and achieve significant efficiency gains without sacrificing performance. Our code is publicly available at https://github.com/RaghavSinghal10/lora-sb.

Diffusion models achieve state-of-the-art performance in various generation tasks. However, their theoretical foundations fall far behind. This paper studies score approximation, estimation, and distribution recovery of diffusion models, when data are supported on an unknown low-dimensional linear subspace. Our result provides sample complexity bounds for distribution estimation using diffusion models. We show that with a properly chosen neural network architecture, the score function can be both accurately approximated and efficiently estimated. Furthermore, the generated distribution based on the estimated score function captures the data geometric structures and converges to a close vicinity of the data distribution. The convergence rate depends on the subspace dimension, indicating that diffusion models can circumvent the curse of data ambient dimensionality.

In large scale machine learning, random sampling is a popular way to approximate datasets by a small representative subset of examples. In particular, sensitivity sampling is an intensely studied technique which provides provable guarantees on the quality of approximation, while reducing the number of examples to the product of the VC dimension d and the total sensitivity mathfrak S in remarkably general settings. However, guarantees going beyond this general bound of mathfrak S d are known in perhaps only one setting, for ell_2 subspace embeddings, despite intense study of sensitivity sampling in prior work. In this work, we show the first bounds for sensitivity sampling for ell_p subspace embeddings for pneq 2 that improve over the general mathfrak S d bound, achieving a bound of roughly mathfrak S^{2/p} for 1leq p<2 and mathfrak S^{2-2/p} for 2<p<infty. For 1leq p<2, we show that this bound is tight, in the sense that there exist matrices for which mathfrak S^{2/p} samples is necessary. Furthermore, our techniques yield further new results in the study of sampling algorithms, showing that the root leverage score sampling algorithm achieves a bound of roughly d for 1leq p<2, and that a combination of leverage score and sensitivity sampling achieves an improved bound of roughly d^{2/p}mathfrak S^{2-4/p} for 2<p<infty. Our sensitivity sampling results yield the best known sample complexity for a wide class of structured matrices that have small ell_p sensitivity.

Shape generation is the practice of producing 3D shapes as various representations for 3D content creation. Previous studies on 3D shape generation have focused on shape quality and structure, without or less considering the importance of semantic information. Consequently, such generative models often fail to preserve the semantic consistency of shape structure or enable manipulation of the semantic attributes of shapes during generation. In this paper, we proposed a novel semantic generative model named 3D Semantic Subspace Traverser that utilizes semantic attributes for category-specific 3D shape generation and editing. Our method utilizes implicit functions as the 3D shape representation and combines a novel latent-space GAN with a linear subspace model to discover semantic dimensions in the local latent space of 3D shapes. Each dimension of the subspace corresponds to a particular semantic attribute, and we can edit the attributes of generated shapes by traversing the coefficients of those dimensions. Experimental results demonstrate that our method can produce plausible shapes with complex structures and enable the editing of semantic attributes. The code and trained models are available at https://github.com/TrepangCat/3D_Semantic_Subspace_Traverser

Subspace representation is a fundamental technique in various fields of machine learning. Analyzing a geometrical relationship among multiple subspaces is essential for understanding subspace series' temporal and/or spatial dynamics. This paper proposes the second-order difference subspace, a higher-order extension of the first-order difference subspace between two subspaces that can analyze the geometrical difference between them. As a preliminary for that, we extend the definition of the first-order difference subspace to the more general setting that two subspaces with different dimensions have an intersection. We then define the second-order difference subspace by combining the concept of first-order difference subspace and principal component subspace (Karcher mean) between two subspaces, motivated by the second-order central difference method. We can understand that the first/second-order difference subspaces correspond to the velocity and acceleration of subspace dynamics from the viewpoint of a geodesic on a Grassmann manifold. We demonstrate the validity and naturalness of our second-order difference subspace by showing numerical results on two applications: temporal shape analysis of a 3D object and time series analysis of a biometric signal.

Recent empirical studies have demonstrated that diffusion models can effectively learn the image distribution and generate new samples. Remarkably, these models can achieve this even with a small number of training samples despite a large image dimension, circumventing the curse of dimensionality. In this work, we provide theoretical insights into this phenomenon by leveraging key empirical observations: (i) the low intrinsic dimensionality of image data, (ii) a union of manifold structure of image data, and (iii) the low-rank property of the denoising autoencoder in trained diffusion models. These observations motivate us to assume the underlying data distribution of image data as a mixture of low-rank Gaussians and to parameterize the denoising autoencoder as a low-rank model according to the score function of the assumed distribution. With these setups, we rigorously show that optimizing the training loss of diffusion models is equivalent to solving the canonical subspace clustering problem over the training samples. Based on this equivalence, we further show that the minimal number of samples required to learn the underlying distribution scales linearly with the intrinsic dimensions under the above data and model assumptions. This insight sheds light on why diffusion models can break the curse of dimensionality and exhibit the phase transition in learning distributions. Moreover, we empirically establish a correspondence between the subspaces and the semantic representations of image data, facilitating image editing. We validate these results with corroborated experimental results on both simulated distributions and image datasets.

Input space reconstruction is an attractive representation learning paradigm. Despite interpretability of the reconstruction and generation, we identify a misalignment between learning by reconstruction, and learning for perception. We show that the former allocates a model's capacity towards a subspace of the data explaining the observed variance--a subspace with uninformative features for the latter. For example, the supervised TinyImagenet task with images projected onto the top subspace explaining 90\% of the pixel variance can be solved with 45\% test accuracy. Using the bottom subspace instead, accounting for only 20\% of the pixel variance, reaches 55\% test accuracy. The features for perception being learned last explains the need for long training time, e.g., with Masked Autoencoders. Learning by denoising is a popular strategy to alleviate that misalignment. We prove that while some noise strategies such as masking are indeed beneficial, others such as additive Gaussian noise are not. Yet, even in the case of masking, we find that the benefits vary as a function of the mask's shape, ratio, and the considered dataset. While tuning the noise strategy without knowledge of the perception task seems challenging, we provide first clues on how to detect if a noise strategy is never beneficial regardless of the perception task.

The push to train ever larger neural networks has motivated the study of initialization and training at large network width. A key challenge is to scale training so that a network's internal representations evolve nontrivially at all widths, a process known as feature learning. Here, we show that feature learning is achieved by scaling the spectral norm of weight matrices and their updates like texttt{fan-out/fan-in}, in contrast to widely used but heuristic scalings based on Frobenius norm and entry size. Our spectral scaling analysis also leads to an elementary derivation of maximal update parametrization. All in all, we aim to provide the reader with a solid conceptual understanding of feature learning in neural networks.

Despite large neural networks demonstrating remarkable abilities to complete different tasks, they require excessive memory usage to store the optimization states for training. To alleviate this, the low-rank adaptation (LoRA) is proposed to reduce the optimization states by training fewer parameters. However, LoRA restricts overall weight update matrices to be low-rank, limiting the model performance. In this work, we investigate the dynamics of LoRA and identify that it can be approximated by a random projection. Based on this observation, we propose Flora, which is able to achieve high-rank updates by resampling the projection matrices while enjoying the sublinear space complexity of optimization states. We conduct experiments across different tasks and model architectures to verify the effectiveness of our approach.

Modern neural models trained on textual data rely on pre-trained representations that emerge without direct supervision. As these representations are increasingly being used in real-world applications, the inability to control their content becomes an increasingly important problem. We formulate the problem of identifying and erasing a linear subspace that corresponds to a given concept, in order to prevent linear predictors from recovering the concept. We model this problem as a constrained, linear maximin game, and show that existing solutions are generally not optimal for this task. We derive a closed-form solution for certain objectives, and propose a convex relaxation, \method, that works well for others. When evaluated in the context of binary gender removal, the method recovers a low-dimensional subspace whose removal mitigates bias by intrinsic and extrinsic evaluation. We show that the method is highly expressive, effectively mitigating bias in deep nonlinear classifiers while maintaining tractability and interpretability.

Large-scale text-to-image diffusion models excel in generating high-quality images from textual inputs, yet concerns arise as research indicates their tendency to memorize and replicate training data, raising We also addressed the issue of memorization in diffusion models, where models tend to replicate exact training samples raising copyright infringement and privacy issues. Efforts within the text-to-image community to address memorization explore causes such as data duplication, replicated captions, or trigger tokens, proposing per-prompt inference-time or training-time mitigation strategies. In this paper, we focus on the feed-forward layers and begin by contrasting neuron activations of a set of memorized and non-memorized prompts. Experiments reveal a surprising finding: many different sets of memorized prompts significantly activate a common subspace in the model, demonstrating, for the first time, that memorization in the diffusion models lies in a special subspace. Subsequently, we introduce a novel post-hoc method for editing pre-trained models, whereby memorization is mitigated through the straightforward pruning of weights in specialized subspaces, avoiding the need to disrupt the training or inference process as seen in prior research. Finally, we demonstrate the robustness of the pruned model against training data extraction attacks, thereby unveiling new avenues for a practical and one-for-all solution to memorization.

We show that deep neural networks trained across diverse tasks exhibit remarkably similar low-dimensional parametric subspaces. We provide the first large-scale empirical evidence that demonstrates that neural networks systematically converge to shared spectral subspaces regardless of initialization, task, or domain. Through mode-wise spectral analysis of over 1100 models - including 500 Mistral-7B LoRAs, 500 Vision Transformers, and 50 LLaMA-8B models - we identify universal subspaces capturing majority variance in just a few principal directions. By applying spectral decomposition techniques to the weight matrices of various architectures trained on a wide range of tasks and datasets, we identify sparse, joint subspaces that are consistently exploited, within shared architectures across diverse tasks and datasets. Our findings offer new insights into the intrinsic organization of information within deep networks and raise important questions about the possibility of discovering these universal subspaces without the need for extensive data and computational resources. Furthermore, this inherent structure has significant implications for model reusability, multi-task learning, model merging, and the development of training and inference-efficient algorithms, potentially reducing the carbon footprint of large-scale neural models.

Post-training quantization (PTQ) of large language models (LLMs) holds the promise in reducing the prohibitive computational cost at inference time. Quantization of all weight, activation and key-value (KV) cache tensors to 4-bit without significantly degrading generalizability is challenging, due to the high quantization error caused by extreme outliers in activations. To tackle this problem, we propose ResQ, a PTQ method that pushes further the state-of-the-art. By means of principal component analysis (PCA), it identifies a low-rank subspace (in practice 1/8 of the hidden dimension) in which activation variances are highest, and keep the coefficients within this subspace in high precision, e.g. 8-bit, while quantizing the rest to 4-bit. Within each subspace, invariant random rotation is applied to further suppress outliers. We show that this is a provably optimal mixed precision quantization scheme that minimizes error. With the Llama and Qwen2.5 families of models, we demonstrate that ResQ outperforms recent uniform and mixed precision PTQ methods on a variety of benchmarks, achieving up to 33\% lower perplexity on Wikitext than the next best method SpinQuant, and upto 3\times speedup over 16-bit baseline. Code is available at https://github.com/utkarsh-dmx/project-resq.

In this paper, we introduce a subspace-inspired Low-Rank Adaptation (LoRA) method, which is computationally efficient, easy to implement, and readily applicable to large language, multimodal, and diffusion models. Initially, we equivalently decompose the weights of LoRA into two subspaces, and find that simply mixing them can enhance performance. To study such a phenomenon, we revisit it through a fine-grained subspace lens, showing that such modification is equivalent to employing a fixed mixer to fuse the subspaces. To be more flexible, we jointly learn the mixer with the original LoRA weights, and term the method Mixture-of-Subspaces LoRA (MoSLoRA). MoSLoRA consistently outperforms LoRA on tasks in different modalities, including commonsense reasoning, visual instruction tuning, and subject-driven text-to-image generation, demonstrating its effectiveness and robustness. Codes are available at https://github.com/wutaiqiang/MoSLoRA{github}.

During typical gradient-based training of deep neural networks, all of the model's parameters are updated at each iteration. Recent work has shown that it is possible to update only a small subset of the model's parameters during training, which can alleviate storage and communication requirements. In this paper, we show that it is possible to induce a fixed sparse mask on the model's parameters that selects a subset to update over many iterations. Our method constructs the mask out of the k parameters with the largest Fisher information as a simple approximation as to which parameters are most important for the task at hand. In experiments on parameter-efficient transfer learning and distributed training, we show that our approach matches or exceeds the performance of other methods for training with sparse updates while being more efficient in terms of memory usage and communication costs. We release our code publicly to promote further applications of our approach.

Recent works have shown that, when trained at scale, uni-modal 2D vision and text encoders converge to learned features that share remarkable structural properties, despite arising from different representations. However, the role of 3D encoders with respect to other modalities remains unexplored. Furthermore, existing 3D foundation models that leverage large datasets are typically trained with explicit alignment objectives with respect to frozen encoders from other representations. In this work, we investigate the possibility of a posteriori alignment of representations obtained from uni-modal 3D encoders compared to text-based feature spaces. We show that naive post-training feature alignment of uni-modal text and 3D encoders results in limited performance. We then focus on extracting subspaces of the corresponding feature spaces and discover that by projecting learned representations onto well-chosen lower-dimensional subspaces the quality of alignment becomes significantly higher, leading to improved accuracy on matching and retrieval tasks. Our analysis further sheds light on the nature of these shared subspaces, which roughly separate between semantic and geometric data representations. Overall, ours is the first work that helps to establish a baseline for post-training alignment of 3D uni-modal and text feature spaces, and helps to highlight both the shared and unique properties of 3D data compared to other representations.

Low-rank approximation is a fundamental technique in modern data analysis, widely utilized across various fields such as signal processing, machine learning, and natural language processing. Despite its ubiquity, the mechanics of low-rank approximation and its application in adaptation can sometimes be obscure, leaving practitioners and researchers with questions about its true capabilities and limitations. This paper seeks to clarify low-rank approximation and adaptation by offering a comprehensive guide that reveals their inner workings and explains their utility in a clear and accessible way. Our focus here is to develop a solid intuition for how low-rank approximation and adaptation operate, and why they are so effective. We begin with basic concepts and gradually build up to the mathematical underpinnings, ensuring that readers of all backgrounds can gain a deeper understanding of low-rank approximation and adaptation. We strive to strike a balance between informal explanations and rigorous mathematics, ensuring that both newcomers and experienced experts can benefit from this survey. Additionally, we introduce new low-rank decomposition and adaptation algorithms that have not yet been explored in the field, hoping that future researchers will investigate their potential applicability.

Maintaining an up-to-date map that accurately reflects recent changes in the environment is crucial, especially for robots that repeatedly traverse the same space. Failing to promptly update the changed regions can degrade map quality, resulting in poor localization, inefficient operations, and even lost robots. 3D Gaussian Splatting (3DGS) has recently seen widespread adoption in online map reconstruction due to its dense, differentiable, and photorealistic properties, yet accurately and efficiently updating the regions of change remains a challenge. In this paper, we propose VG-Mapping, a novel online 3DGS-based mapping system tailored for such semi-static scenes. Our approach introduces a hybrid representation that augments 3DGS with a TSDF-based voxel map to efficiently identify changed regions in a scene, along with a variation-aware density control strategy that inserts or deletes Gaussian primitives in regions undergoing change. Furthermore, to address the absence of public benchmarks for this task, we construct a RGB-D dataset comprising both synthetic and real-world semi-static environments. Experimental results demonstrate that our method substantially improves the rendering quality and map update efficiency in semi-static scenes. The code and dataset are available at https://github.com/heyicheng-never/VG-Mapping.

Training modern neural networks on large datasets is computationally and environmentally costly. We introduce GRAFT, a scalable in-training subset selection method that (i) extracts a low-rank feature representation for each batch, (ii) applies a Fast MaxVol sampler to select a small, diverse subset that spans the batch's dominant subspace, and (iii) dynamically adjusts the subset size using a gradient-approximation criterion. By operating in low-rank subspaces and training on carefully chosen examples instead of full batches, GRAFT preserves the training trajectory while reducing wall-clock time, energy consumption, and CO_2 emissions. Across multiple benchmarks, GRAFT matches or exceeds recent selection baselines in both accuracy and efficiency, providing a favorable trade-off between accuracy, efficiency, and emissions.

We introduce Evolution Guided General Optimization via Low-rank Learning (EGGROLL), an evolution strategies (ES) algorithm designed to scale backprop-free optimization to large population sizes for modern large neural network architectures with billions of parameters. ES is a set of powerful blackbox optimisation methods that can handle non-differentiable or noisy objectives with excellent scaling potential through parallelisation. Na{ï}ve ES becomes prohibitively expensive at scale due to the computational and memory costs associated with generating matrix perturbations EinR^{mtimes n} and the batched matrix multiplications needed to compute per-member forward passes. EGGROLL overcomes these bottlenecks by generating random matrices Ain R^{mtimes r}, Bin R^{ntimes r} with rll min(m,n) to form a low-rank matrix perturbation A B^top that are used in place of the full-rank perturbation E. As the overall update is an average across a population of N workers, this still results in a high-rank update but with significant memory and computation savings, reducing the auxiliary storage from mn to r(m+n) per layer and the cost of a forward pass from O(mn) to O(r(m+n)) when compared to full-rank ES. A theoretical analysis reveals our low-rank update converges to the full-rank update at a fast Oleft(1{r}right) rate. Our experiments show that (1) EGGROLL does not compromise the performance of ES in tabula-rasa RL settings, despite being faster, (2) it is competitive with GRPO as a technique for improving LLM reasoning, and (3) EGGROLL enables stable pre-training of nonlinear recurrent language models that operate purely in integer datatypes.

Accurate, up-to-date High-Definition (HD) maps are critical for urban planning, infrastructure monitoring, and autonomous navigation. However, these maps quickly become outdated as environments evolve, creating a need for robust methods that not only detect changes but also incorporate them into updated 3D representations. While change detection techniques have advanced significantly, there remains a clear gap between detecting changes and actually updating 3D maps, particularly when relying on 2D image-based change detection. To address this gap, we introduce SceneEdited, the first city-scale dataset explicitly designed to support research on HD map maintenance through 3D point cloud updating. SceneEdited contains over 800 up-to-date scenes covering 73 km of driving and approximate 3 km^2 of urban area, with more than 23,000 synthesized object changes created both manually and automatically across 2000+ out-of-date versions, simulating realistic urban modifications such as missing roadside infrastructure, buildings, overpasses, and utility poles. Each scene includes calibrated RGB images, LiDAR scans, and detailed change masks for training and evaluation. We also provide baseline methods using a foundational image-based structure-from-motion pipeline for updating outdated scenes, as well as a comprehensive toolkit supporting scalability, trackability, and portability for future dataset expansion and unification of out-of-date object annotations. Both the dataset and the toolkit are publicly available at https://github.com/ChadLin9596/ScenePoint-ETK, establising a standardized benchmark for 3D map updating research.

Bayesian neural networks (BNNs) offer uncertainty quantification but come with the downside of substantially increased training and inference costs. Sparse BNNs have been investigated for efficient inference, typically by either slowly introducing sparsity throughout the training or by post-training compression of dense BNNs. The dilemma of how to cut down massive training costs remains, particularly given the requirement to learn about the uncertainty. To solve this challenge, we introduce Sparse Subspace Variational Inference (SSVI), the first fully sparse BNN framework that maintains a consistently highly sparse Bayesian model throughout the training and inference phases. Starting from a randomly initialized low-dimensional sparse subspace, our approach alternately optimizes the sparse subspace basis selection and its associated parameters. While basis selection is characterized as a non-differentiable problem, we approximate the optimal solution with a removal-and-addition strategy, guided by novel criteria based on weight distribution statistics. Our extensive experiments show that SSVI sets new benchmarks in crafting sparse BNNs, achieving, for instance, a 10-20x compression in model size with under 3\% performance drop, and up to 20x FLOPs reduction during training compared with dense VI training. Remarkably, SSVI also demonstrates enhanced robustness to hyperparameters, reducing the need for intricate tuning in VI and occasionally even surpassing VI-trained dense BNNs on both accuracy and uncertainty metrics.

The rapid growth of large models has raised concerns about their environmental impact and equity in accessibility due to significant computational costs. Low-Rank Adapters (LoRA) offer a lightweight solution for finetuning large models, resulting in an abundance of publicly available adapters tailored to diverse domains. We ask: Can these pretrained adapters be leveraged to further streamline adaptation to new tasks while addressing these challenges? We introduce EigenLoRAx, a parameter-efficient finetuning method that recycles existing adapters to create a principal subspace aligned with their shared domain knowledge which can be further augmented with orthogonal basis vectors in low-resource scenarios. This enables rapid adaptation to new tasks by learning only lightweight coefficients on the principal components of the subspace - eliminating the need to finetune entire adapters. EigenLoRAx requires significantly fewer parameters and memory, improving efficiency for both training and inference. Our method demonstrates strong performance across diverse domains and tasks, offering a scalable for edge-based applications, personalization, and equitable deployment of large models in resource-constrained environments.

Neural networks have achieved tremendous success in a large variety of applications. However, their memory footprint and computational demand can render them impractical in application settings with limited hardware or energy resources. In this work, we propose a novel algorithm to find efficient low-rank subnetworks. Remarkably, these subnetworks are determined and adapted already during the training phase and the overall time and memory resources required by both training and evaluating them are significantly reduced. The main idea is to restrict the weight matrices to a low-rank manifold and to update the low-rank factors rather than the full matrix during training. To derive training updates that are restricted to the prescribed manifold, we employ techniques from dynamic model order reduction for matrix differential equations. This allows us to provide approximation, stability, and descent guarantees. Moreover, our method automatically and dynamically adapts the ranks during training to achieve the desired approximation accuracy. The efficiency of the proposed method is demonstrated through a variety of numerical experiments on fully-connected and convolutional networks.

Deep subspace clustering (DSC) networks based on self-expressive model learn representation matrix, often implemented in terms of fully connected network, in the embedded space. After the learning is finished, representation matrix is used by spectral clustering module to assign labels to clusters. However, such approach ignores complementary information that exist in other layers of the encoder (including the input data themselves). Herein, we apply selected linear subspace clustering algorithm to learn representation matrices from representations learned by all layers of encoder network including the input data. Afterward, we learn a multilayer graph that in a multi-view like manner integrates information from graph Laplacians of all used layers. That improves further performance of selected DSC network. Furthermore, we also provide formulation of our approach to cluster out-of-sample/test data points. We validate proposed approach on four well-known datasets with two DSC networks as baseline models. In almost all the cases, proposed approach achieved statistically significant improvement in three performance metrics. MATLAB code of proposed algorithm is posted on https://github.com/lovro-sinda/MLG-DSC.

Model merging aims to cheaply combine individual task-specific models into a single multitask model. In this work, we view past merging methods as leveraging different notions of a ''task subspace'' in which models are matched before being merged. We connect the task subspace of a given model to its loss landscape and formalize how this approach to model merging can be seen as solving a linear system of equations. While past work has generally been limited to linear systems that have a closed-form solution, we consider using the conjugate gradient method to find a solution. We show that using the conjugate gradient method can outperform closed-form solutions, enables merging via linear systems that are otherwise intractable to solve, and flexibly allows choosing from a wide variety of initializations and estimates for the ''task subspace''. We ultimately demonstrate that our merging framework called ''Matching Models in their Task Subspace'' (MaTS) achieves state-of-the-art results in multitask and intermediate-task model merging. We release all of the code and checkpoints used in our work at https://github.com/r-three/mats.

LoRA-based large model parameter-efficient fine-tuning (PEFT) methods use low-rank de- composition to approximate updates to model parameters. However, compared to full- parameter fine-tuning, low-rank updates often lead to a performance gap in downstream tasks. To address this, we introduce DropLoRA, a novel pruning-based approach that focuses on pruning the rank dimension. Unlike conven- tional methods that attempt to overcome the low-rank bottleneck, DropLoRA innovatively integrates a pruning module between the two low-rank matrices in LoRA to simulate dy- namic subspace learning. This dynamic low- rank subspace learning allows DropLoRA to overcome the limitations of traditional LoRA, which operates within a static subspace. By continuously adapting the learning subspace, DropLoRA significantly boosts performance without incurring additional training or infer- ence costs. Our experimental results demon- strate that DropLoRA consistently outperforms LoRA in fine-tuning the LLaMA series across a wide range of large language model gener- ation tasks, including commonsense reason- ing, mathematical reasoning, code generation, and instruction-following. Our code is avail- able at https://github.com/TayeeChang/DropLoRA.

While transformer models are widely believed to operate in high-dimensional hidden spaces, we show that attention outputs are confined to a surprisingly low-dimensional subspace, where about 60\% of the directions account for 99\% of the variance--a phenomenon that is induced by the attention output projection matrix and consistently observed across diverse model families and datasets. Critically, we find this low-rank structure as a fundamental cause of the prevalent dead feature problem in sparse dictionary learning, where it creates a mismatch between randomly initialized features and the intrinsic geometry of the activation space. Building on this insight, we propose a subspace-constrained training method for sparse autoencoders (SAEs), initializing feature directions into the active subspace of activations. Our approach reduces dead features from 87\% to below 1\% in Attention Output SAEs with 1M features, and can further extend to other sparse dictionary learning methods. Our findings provide both new insights into the geometry of attention and practical tools for improving sparse dictionary learning in large language models.

Popular PEFT methods achieve parameter efficiency by assuming that incremental weight updates are inherently low-rank, which often leads to a performance gap compared to full fine-tuning. While recent methods have attempted to address this limitation, they typically lack sufficient parameter and memory efficiency. We propose VectorFit, an effective and easily deployable approach that adaptively trains the singular vectors and biases of pre-trained weight matrices. We demonstrate that the utilization of structural and transformational characteristics of pre-trained weights enables high-rank updates comparable to those of full fine-tuning. As a result, VectorFit achieves superior performance with 9X less trainable parameters compared to state-of-the-art PEFT methods. Through extensive experiments over 17 datasets spanning diverse language and vision tasks such as natural language understanding and generation, question answering, image classification, and image generation, we exhibit that VectorFit consistently outperforms baselines, even in extremely low-budget scenarios.

The seminal Fast Johnson-Lindenstrauss (Fast JL) transform by Ailon and Chazelle (SICOMP'09) embeds a set of n points in d-dimensional Euclidean space into optimal k=O(varepsilon^{-2} ln n) dimensions, while preserving all pairwise distances to within a factor (1 pm varepsilon). The Fast JL transform supports computing the embedding of a data point in O(d ln d +k ln^2 n) time, where the d ln d term comes from multiplication with a d times d Hadamard matrix and the k ln^2 n term comes from multiplication with a sparse k times d matrix. Despite the Fast JL transform being more than a decade old, it is one of the fastest dimensionality reduction techniques for many tradeoffs between varepsilon, d and n. In this work, we give a surprising new analysis of the Fast JL transform, showing that the k ln^2 n term in the embedding time can be improved to (k ln^2 n)/alpha for an alpha = Omega(min{varepsilon^{-1}ln(1/varepsilon), ln n}). The improvement follows by using an even sparser matrix. We also complement our improved analysis with a lower bound showing that our new analysis is in fact tight.

Self-supervised learning (SSL) has emerged as a powerful paradigm for learning representations without labeled data, often by enforcing invariance to input transformations such as rotations or blurring. Recent studies have highlighted two pivotal properties for effective representations: (i) avoiding dimensional collapse-where the learned features occupy only a low-dimensional subspace, and (ii) enhancing uniformity of the induced distribution. In this work, we introduce T-REGS, a simple regularization framework for SSL based on the length of the Minimum Spanning Tree (MST) over the learned representation. We provide theoretical analysis demonstrating that T-REGS simultaneously mitigates dimensional collapse and promotes distribution uniformity on arbitrary compact Riemannian manifolds. Several experiments on synthetic data and on classical SSL benchmarks validate the effectiveness of our approach at enhancing representation quality.

Personalized text-to-image generation has attracted unprecedented attention in the recent few years due to its unique capability of generating highly-personalized images via using the input concept dataset and novel textual prompt. However, previous methods solely focus on the performance of the reconstruction task, degrading its ability to combine with different textual prompt. Besides, optimizing in the high-dimensional embedding space usually leads to unnecessary time-consuming training process and slow convergence. To address these issues, we propose an efficient method to explore the target embedding in a textual subspace, drawing inspiration from the self-expressiveness property. Additionally, we propose an efficient selection strategy for determining the basis vectors of the textual subspace. The experimental evaluations demonstrate that the learned embedding can not only faithfully reconstruct input image, but also significantly improves its alignment with novel input textual prompt. Furthermore, we observe that optimizing in the textual subspace leads to an significant improvement of the robustness to the initial word, relaxing the constraint that requires users to input the most relevant initial word. Our method opens the door to more efficient representation learning for personalized text-to-image generation.

Foundational mapping remains a challenge in many parts of the world, particularly in dynamic scenarios such as natural disasters when timely updates are critical. Updating maps is currently a highly manual process requiring a large number of human labelers to either create features or rigorously validate automated outputs. We propose that the frequent revisits of earth imaging satellite constellations may accelerate existing efforts to quickly update foundational maps when combined with advanced machine learning techniques. Accordingly, the SpaceNet partners (CosmiQ Works, Radiant Solutions, and NVIDIA), released a large corpus of labeled satellite imagery on Amazon Web Services (AWS) called SpaceNet. The SpaceNet partners also launched a series of public prize competitions to encourage improvement of remote sensing machine learning algorithms. The first two of these competitions focused on automated building footprint extraction, and the most recent challenge focused on road network extraction. In this paper we discuss the SpaceNet imagery, labels, evaluation metrics, prize challenge results to date, and future plans for the SpaceNet challenge series.

Backward compatibility of model predictions is a desired property when updating a machine learning driven application. It allows to seamlessly improve the underlying model without introducing regression bugs. In classification tasks these bugs occur in the form of negative flips. This means an instance that was correctly classified by the old model is now classified incorrectly by the updated model. This has direct negative impact on the user experience of such systems e.g. a frequently used voice assistant query is suddenly misclassified. A common reason to update the model is when new training data becomes available and needs to be incorporated. Simply retraining the model with the updated data introduces the unwanted negative flips. We study the problem of regression during data updates and propose Backward Compatible Weight Interpolation (BCWI). This method interpolates between the weights of the old and new model and we show in extensive experiments that it reduces negative flips without sacrificing the improved accuracy of the new model. BCWI is straight forward to implement and does not increase inference cost. We also explore the use of importance weighting during interpolation and averaging the weights of multiple new models in order to further reduce negative flips.

A major goal of unsupervised learning is to discover data representations that are useful for subsequent tasks, without access to supervised labels during training. Typically, this involves minimizing a surrogate objective, such as the negative log likelihood of a generative model, with the hope that representations useful for subsequent tasks will arise as a side effect. In this work, we propose instead to directly target later desired tasks by meta-learning an unsupervised learning rule which leads to representations useful for those tasks. Specifically, we target semi-supervised classification performance, and we meta-learn an algorithm -- an unsupervised weight update rule -- that produces representations useful for this task. Additionally, we constrain our unsupervised update rule to a be a biologically-motivated, neuron-local function, which enables it to generalize to different neural network architectures, datasets, and data modalities. We show that the meta-learned update rule produces useful features and sometimes outperforms existing unsupervised learning techniques. We further show that the meta-learned unsupervised update rule generalizes to train networks with different widths, depths, and nonlinearities. It also generalizes to train on data with randomly permuted input dimensions and even generalizes from image datasets to a text task.

This paper investigates the generalization of Principal Component Analysis (PCA) to Riemannian manifolds. We first propose a new and general type of family of subspaces in manifolds that we call barycentric subspaces. They are implicitly defined as the locus of points which are weighted means of k+1 reference points. As this definition relies on points and not on tangent vectors, it can also be extended to geodesic spaces which are not Riemannian. For instance, in stratified spaces, it naturally allows principal subspaces that span several strata, which is impossible in previous generalizations of PCA. We show that barycentric subspaces locally define a submanifold of dimension k which generalizes geodesic subspaces.Second, we rephrase PCA in Euclidean spaces as an optimization on flags of linear subspaces (a hierarchy of properly embedded linear subspaces of increasing dimension). We show that the Euclidean PCA minimizes the Accumulated Unexplained Variances by all the subspaces of the flag (AUV). Barycentric subspaces are naturally nested, allowing the construction of hierarchically nested subspaces. Optimizing the AUV criterion to optimally approximate data points with flags of affine spans in Riemannian manifolds lead to a particularly appealing generalization of PCA on manifolds called Barycentric Subspaces Analysis (BSA).

Growing amount and quality of AI-generated texts makes detecting such content more difficult. In most real-world scenarios, the domain (style and topic) of generated data and the generator model are not known in advance. In this work, we focus on the robustness of classifier-based detectors of AI-generated text, namely their ability to transfer to unseen generators or semantic domains. We investigate the geometry of the embedding space of Transformer-based text encoders and show that clearing out harmful linear subspaces helps to train a robust classifier, ignoring domain-specific spurious features. We investigate several subspace decomposition and feature selection strategies and achieve significant improvements over state of the art methods in cross-domain and cross-generator transfer. Our best approaches for head-wise and coordinate-based subspace removal increase the mean out-of-distribution (OOD) classification score by up to 9% and 14% in particular setups for RoBERTa and BERT embeddings respectively. We release our code and data: https://github.com/SilverSolver/RobustATD

The sufficiently scattered condition (SSC) is a key condition in the study of identifiability of various matrix factorization problems, including nonnegative, minimum-volume, symmetric, simplex-structured, and polytopic matrix factorizations. The SSC allows one to guarantee that the computed matrix factorization is unique/identifiable, up to trivial ambiguities. However, this condition is NP-hard to check in general. In this paper, we show that it can however be checked in a reasonable amount of time in realistic scenarios, when the factorization rank is not too large. This is achieved by formulating the problem as a non-convex quadratic optimization problem over a bounded set. We use the global non-convex optimization software Gurobi, and showcase the usefulness of this code on synthetic data sets and on real-world hyperspectral images.

The Maximal Update Parametrization (muP) aims to make the optimal hyperparameters (HPs) of a model independent of its size, allowing them to be swept using a cheap proxy model rather than the full-size target model. We present a new scheme, u-muP, which improves upon muP by combining it with Unit Scaling, a method for designing models that makes them easy to train in low-precision. The two techniques have a natural affinity: muP ensures that the scale of activations is independent of model size, and Unit Scaling ensures that activations, weights and gradients begin training with a scale of one. This synthesis opens the door to a simpler scheme, whose default values are near-optimal. This in turn facilitates a more efficient sweeping strategy, with u-muP models reaching a lower loss than comparable muP models and working out-of-the-box in FP8.

Semidefinite programming is a fundamental problem class in convex optimization, but despite recent advances in solvers, solving large-scale semidefinite programs remains challenging. Generally the matrix functions involved are spectral or unitarily invariant, i.e., they depend only on the eigenvalues or singular values of the matrix. This paper investigates how spectral matrix cones -- cones defined from epigraphs and perspectives of spectral or unitarily invariant functions -- can be used to enhance first-order conic solvers for semidefinite programs. Our main result shows that projecting a matrix can be reduced to projecting its eigenvalues or singular values, which we demonstrate can be done at a negligible cost compared to the eigenvalue or singular value decomposition itself. We have integrated support for spectral matrix cone projections into the Splitting Conic Solver (SCS). Numerical experiments show that SCS with this enhancement can achieve speedups of up to an order of magnitude for solving semidefinite programs arising in experimental design, robust principal component analysis, and graph partitioning.

Vector Symbolic Architectures (VSAs) have emerged as a novel framework for enabling interpretable machine learning algorithms equipped with the ability to reason and explain their decision processes. The basic idea is to represent discrete information through high dimensional random vectors. Complex data structures can be built up with operations over vectors such as the "binding" operation involving element-wise vector multiplication, which associates data together. The reverse task of decomposing the associated elements is a combinatorially hard task, with an exponentially large search space. The main algorithm for performing this search is the resonator network, inspired by Hopfield network-based memory search operations. In this work, we introduce a new variant of the resonator network, based on self-attention based update rules in the iterative search problem. This update rule, based on the Hopfield network with log-sum-exp energy function and norm-bounded states, is shown to substantially improve the performance and rate of convergence. As a result, our algorithm enables a larger capacity for associative memory, enabling applications in many tasks like perception based pattern recognition, scene decomposition, and object reasoning. We substantiate our algorithm with a thorough evaluation and comparisons to baselines.

Large pre-trained models (LPMs) have demonstrated exceptional performance in diverse natural language processing and computer vision tasks. However, fully fine-tuning these models poses substantial memory challenges, particularly in resource-constrained environments. Parameter-efficient fine-tuning (PEFT) methods, such as LoRA, mitigate this issue by adjusting only a small subset of parameters. Nevertheless, these methods typically employ random initialization for low-rank matrices, which can lead to inefficiencies in gradient descent and diminished generalizability due to suboptimal starting points. To address these limitations, we propose SVFit, a novel PEFT approach that leverages singular value decomposition (SVD) to initialize low-rank matrices using critical singular values as trainable parameters. Specifically, SVFit performs SVD on the pre-trained weight matrix to obtain the best rank-r approximation matrix, emphasizing the most critical singular values that capture over 99% of the matrix's information. These top-r singular values are then used as trainable parameters to scale the fundamental subspaces of the matrix, facilitating rapid domain adaptation. Extensive experiments across various pre-trained models in natural language understanding, text-to-image generation, and image classification tasks reveal that SVFit outperforms LoRA while requiring 16 times fewer trainable parameters.

We present EGN, a stochastic second-order optimization algorithm that combines the generalized Gauss-Newton (GN) Hessian approximation with low-rank linear algebra to compute the descent direction. Leveraging the Duncan-Guttman matrix identity, the parameter update is obtained by factorizing a matrix which has the size of the mini-batch. This is particularly advantageous for large-scale machine learning problems where the dimension of the neural network parameter vector is several orders of magnitude larger than the batch size. Additionally, we show how improvements such as line search, adaptive regularization, and momentum can be seamlessly added to EGN to further accelerate the algorithm. Moreover, under mild assumptions, we prove that our algorithm converges to an epsilon-stationary point at a linear rate. Finally, our numerical experiments demonstrate that EGN consistently exceeds, or at most matches the generalization performance of well-tuned SGD, Adam, and SGN optimizers across various supervised and reinforcement learning tasks.

By interpreting the product of the Principal Component Analysis, that is the covariance matrix, as a sequence of nested subspaces naturally coming with weights according to the level of approximation they provide, we are able to embed all d--dimensional Grassmannians into a stratified space of covariance matrices. We observe that Grassmannians constitute the lowest dimensional skeleton of the stratification while it is possible to define a Riemaniann metric on the highest dimensional and dense stratum, such a metric being compatible with the global stratification. With such a Riemaniann metric at hand, it is possible to look for geodesics between two linear subspaces of different dimensions that do not go through higher dimensional linear subspaces as would euclidean geodesics. Building upon the proposed embedding of Grassmannians into the stratified space of covariance matrices, we generalize the concept of varifolds to what we call flagfolds in order to model multi-dimensional shapes.

Recently, State Space Models (SSMs), with Mamba as a prime example, have shown great promise for long-range dependency modeling with linear complexity. Then, Vision Mamba and the subsequent architectures are presented successively, and they perform well on visual tasks. The crucial step of applying Mamba to visual tasks is to construct 2D visual features in sequential manners. To effectively organize and construct visual features within the 2D image space through 1D selective scan, we propose a novel Multi-Head Scan (MHS) module. The embeddings extracted from the preceding layer are projected into multiple lower-dimensional subspaces. Subsequently, within each subspace, the selective scan is performed along distinct scan routes. The resulting sub-embeddings, obtained from the multi-head scan process, are then integrated and ultimately projected back into the high-dimensional space. Moreover, we incorporate a Scan Route Attention (SRA) mechanism to enhance the module's capability to discern complex structures. To validate the efficacy of our module, we exclusively substitute the 2D-Selective-Scan (SS2D) block in VM-UNet with our proposed module, and we train our models from scratch without using any pre-trained weights. The results indicate a significant improvement in performance while reducing the parameters of the original VM-UNet. The code for this study is publicly available at https://github.com/PixDeep/MHS-VM.

This paper describes a new technique for finding disentangled semantic directions in the latent space of StyleGAN. Our method identifies meaningful orthogonal subspaces that allow editing of one human face attribute, while minimizing undesired changes in other attributes. Our model is capable of editing a single attribute in multiple directions, resulting in a range of possible generated images. We compare our scheme with three state-of-the-art models and show that our method outperforms them in terms of face editing and disentanglement capabilities. Additionally, we suggest quantitative measures for evaluating attribute separation and disentanglement, and exhibit the superiority of our model with respect to those measures.

Adapting large-scale pre-trained generative models in a parameter-efficient manner is gaining traction. Traditional methods like low rank adaptation achieve parameter efficiency by imposing constraints but may not be optimal for tasks requiring high representation capacity. We propose a novel spectrum-aware adaptation framework for generative models. Our method adjusts both singular values and their basis vectors of pretrained weights. Using the Kronecker product and efficient Stiefel optimizers, we achieve parameter-efficient adaptation of orthogonal matrices. We introduce Spectral Orthogonal Decomposition Adaptation (SODA), which balances computational efficiency and representation capacity. Extensive evaluations on text-to-image diffusion models demonstrate SODA's effectiveness, offering a spectrum-aware alternative to existing fine-tuning methods.

In recent years, diffusion models have become the most popular and powerful methods in the field of image synthesis, even rivaling human artists in artistic creativity. However, the key issue currently limiting the application of diffusion models is its extremely slow generation process. Although several methods were proposed to speed up the generation process, there still exists a trade-off between efficiency and quality. In this paper, we first provide a detailed theoretical and empirical analysis of the generation process of the diffusion models based on schedulers. We transform the designing problem of schedulers into the determination of several parameters, and further transform the accelerated generation process into an expansion process of the linear subspace. Based on these analyses, we consequently propose a novel method called Optimal Linear Subspace Search (OLSS), which accelerates the generation process by searching for the optimal approximation process of the complete generation process in the linear subspaces spanned by latent variables. OLSS is able to generate high-quality images with a very small number of steps. To demonstrate the effectiveness of our method, we conduct extensive comparative experiments on open-source diffusion models. Experimental results show that with a given number of steps, OLSS can significantly improve the quality of generated images. Using an NVIDIA A100 GPU, we make it possible to generate a high-quality image by Stable Diffusion within only one second without other optimization techniques.

Recent work has shown that orthonormal matrix updates speed up neural network optimization, improve training stability, and offer better hyperparameter transfer across model sizes. Applying these updates efficiently when model weights and optimizer states are sharded across a large-scale distributed LLM training system remains a major challenge. We introduce Dion (DIstributed OrthoNormalization), a scalable and communication-efficient orthonormalizing optimizer. Dion leverages low-rank approximation and decoupled momentum buffers, eliminating the need for full gradient synchronization while producing numerically equivalent results. It is compatible with simultaneous DDP, FSDP, and TP parallelism, and it computes an orthonormalized update without unsharding a full parameter matrix on any single device. We evaluate Dion on language models from 120M to 3B parameters and find that its benefits improve with increasing model size and batch size.

We introduce a unified framework for iterative reasoning that leverages non-Euclidean geometry via Bregman divergences, higher-order operator averaging, and adaptive feedback mechanisms. Our analysis establishes that, under mild smoothness and contractivity assumptions, a generalized update scheme not only unifies classical methods such as mirror descent and dynamic programming but also captures modern chain-of-thought reasoning processes in large language models. In particular, we prove that our accelerated iterative update achieves an O(1/t^2) convergence rate in the absence of persistent perturbations, and we further demonstrate that feedback (iterative) architectures are necessary to approximate certain fixed-point functions efficiently. These theoretical insights bridge classical acceleration techniques with contemporary applications in neural computation and optimization.

This work presents a fast and scalable algorithm for incremental learning of Gaussian mixture models. By performing rank-one updates on its precision matrices and determinants, its asymptotic time complexity is of NKD^2 for N data points, K Gaussian components and D dimensions. The resulting algorithm can be applied to high dimensional tasks, and this is confirmed by applying it to the classification datasets MNIST and CIFAR-10. Additionally, in order to show the algorithm's applicability to function approximation and control tasks, it is applied to three reinforcement learning tasks and its data-efficiency is evaluated.

Representations learned via self-supervised learning (SSL) can be susceptible to dimensional collapse, where the learned representation subspace is of extremely low dimensionality and thus fails to represent the full data distribution and modalities. Dimensional collapse also known as the "underfilling" phenomenon is one of the major causes of degraded performance on downstream tasks. Previous work has investigated the dimensional collapse problem of SSL at a global level. In this paper, we demonstrate that representations can span over high dimensional space globally, but collapse locally. To address this, we propose a method called local dimensionality regularization (LDReg). Our formulation is based on the derivation of the Fisher-Rao metric to compare and optimize local distance distributions at an asymptotically small radius for each data point. By increasing the local intrinsic dimensionality, we demonstrate through a range of experiments that LDReg improves the representation quality of SSL. The results also show that LDReg can regularize dimensionality at both local and global levels.

The Koopman operator provides a powerful framework for modeling dynamical systems and has attracted growing interest from the machine learning community. However, its infinite-dimensional nature makes identifying suitable finite-dimensional subspaces challenging, especially for deep architectures. We argue that these difficulties come from suboptimal representation learning, where latent variables fail to balance expressivity and simplicity. This tension is closely related to the information bottleneck (IB) dilemma: constructing compressed representations that are both compact and predictive. Rethinking Koopman learning through this lens, we demonstrate that latent mutual information promotes simplicity, yet an overemphasis on simplicity may cause latent space to collapse onto a few dominant modes. In contrast, expressiveness is sustained by the von Neumann entropy, which prevents such collapse and encourages mode diversity. This insight leads us to propose an information-theoretic Lagrangian formulation that explicitly balances this tradeoff. Furthermore, we propose a new algorithm based on the Lagrangian formulation that encourages both simplicity and expressiveness, leading to a stable and interpretable Koopman representation. Beyond quantitative evaluations, we further visualize the learned manifolds under our representations, observing empirical results consistent with our theoretical predictions. Finally, we validate our approach across a diverse range of dynamical systems, demonstrating improved performance over existing Koopman learning methods. The implementation is publicly available at https://github.com/Wenxuan52/InformationKoopman.

We derive a simple and model-independent formula for the change in the generalization gap due to a gradient descent update. We then compare the change in the test error for stochastic gradient descent to the change in test error from an equivalent number of gradient descent updates and show explicitly that stochastic gradient descent acts to regularize generalization error by decorrelating nearby updates. These calculations depends on the details of the model only through the mean and covariance of the gradient distribution, which may be readily measured for particular models of interest. We discuss further improvements to these calculations and comment on possible implications for stochastic optimization.

Remote sensing change detection aims to localize semantic changes between images of the same location captured at different times. In the past few years, newer methods have attributed enhanced performance to the additions of new and complex components to existing architectures. Most fail to measure the performance contribution of fundamental design choices such as backbone selection, pre-training strategies, and training configurations. We claim that such fundamental design choices often improve performance even more significantly than the addition of new architectural components. Due to that, we systematically revisit the design space of change detection models and analyse the full potential of a well-optimised baseline. We identify a set of fundamental design choices that benefit both new and existing architectures. Leveraging this insight, we demonstrate that when carefully designed, even an architecturally simple model can match or surpass state-of-the-art performance on six challenging change detection datasets. Our best practices generalise beyond our architecture and also offer performance improvements when applied to related methods, indicating that the space of fundamental design choices has been underexplored. Our guidelines and architecture provide a strong foundation for future methods, emphasizing that optimizing core components is just as important as architectural novelty in advancing change detection performance. Code: https://github.com/blaz-r/BTC-change-detection

Large Language Models have shown remarkable capabilities in the NLP domain. Their effectiveness can mainly be attributed to their ability to adapt to an array of downstream tasks. However, generally, full fine-tuning is a computationally expensive job. To mitigate this, many techniques have been developed that prime efficiency, a prominent one being Low-Rank Adaptation (LoRA). However, LoRA and its variants employ re-parametrized additive updates. In this paper, we propose Low-Rank Multiplicative Adaptation (LoRMA), which shifts the paradigm of additive updates to a richer space of matrix multiplicative transformations. We tackle challenges such as computational complexity and rank bottleneck of matrix multiplication by effectively re-ordering operations and introducing rank inflation strategies. We conduct extensive experiments to demonstrate the effectiveness of our approach in terms of various evaluation metrics.

The key-value (KV) cache accelerates LLMs decoding by storing KV tensors from previously generated tokens. It reduces redundant computation at the cost of increased memory usage. To mitigate this overhead, existing approaches compress KV tensors into lower-bit representations; however, quantization errors can accumulate as more tokens are generated, potentially resulting in undesired outputs. In this paper, we introduce SQuat (Subspace-orthogonal KV cache quantization). It first constructs a subspace spanned by query tensors to capture the most critical task-related information. During key tensor quantization, it enforces that the difference between the (de)quantized and original keys remains orthogonal to this subspace, minimizing the impact of quantization errors on the attention mechanism's outputs. SQuat requires no model fine-tuning, no additional calibration dataset for offline learning, and is grounded in a theoretical framework we develop. Through numerical experiments, we show that our method reduces peak memory by 2.17 to 2.82, improves throughput by 2.45 to 3.60, and achieves more favorable benchmark scores than existing KV cache quantization algorithms.

This paper investigates whether large language models (LLMs) utilize numerical attributes encoded in a low-dimensional subspace of the embedding space when answering logical comparison questions (e.g., Was Cristiano born before Messi?). We first identified these subspaces using partial least squares regression, which effectively encodes the numerical attributes associated with the entities in comparison prompts. Further, we demonstrate causality by intervening in these subspaces to manipulate hidden states, thereby altering the LLM's comparison outcomes. Experimental results show that our findings hold for different numerical attributes, indicating that LLMs utilize the linearly encoded information for numerical reasoning.

Modern ML applications increasingly rely on complex deep learning models and large datasets. There has been an exponential growth in the amount of computation needed to train the largest models. Therefore, to scale computation and data, these models are inevitably trained in a distributed manner in clusters of nodes, and their updates are aggregated before being applied to the model. However, a distributed setup is prone to Byzantine failures of individual nodes, components, and software. With data augmentation added to these settings, there is a critical need for robust and efficient aggregation systems. We define the quality of workers as reconstruction ratios in (0,1], and formulate aggregation as a Maximum Likelihood Estimation procedure using Beta densities. We show that the Regularized form of log-likelihood wrt subspace can be approximately solved using iterative least squares solver, and provide convergence guarantees using recent Convex Optimization landscape results. Our empirical findings demonstrate that our approach significantly enhances the robustness of state-of-the-art Byzantine resilient aggregators. We evaluate our method in a distributed setup with a parameter server, and show simultaneous improvements in communication efficiency and accuracy across various tasks. The code is publicly available at https://github.com/hamidralmasi/FlagAggregator

This work studies the combinatorial optimization problem of finding an optimal core tensor shape, also called multilinear rank, for a size-constrained Tucker decomposition. We give an algorithm with provable approximation guarantees for its reconstruction error via connections to higher-order singular values. Specifically, we introduce a novel Tucker packing problem, which we prove is NP-hard, and give a polynomial-time approximation scheme based on a reduction to the 2-dimensional knapsack problem with a matroid constraint. We also generalize our techniques to tree tensor network decompositions. We implement our algorithm using an integer programming solver, and show that its solution quality is competitive with (and sometimes better than) the greedy algorithm that uses the true Tucker decomposition loss at each step, while also running up to 1000x faster.

We rigorously study the joint evolution of training dynamics via stochastic gradient descent (SGD) and the spectra of empirical Hessian and gradient matrices. We prove that in two canonical classification tasks for multi-class high-dimensional mixtures and either 1 or 2-layer neural networks, the SGD trajectory rapidly aligns with emerging low-rank outlier eigenspaces of the Hessian and gradient matrices. Moreover, in multi-layer settings this alignment occurs per layer, with the final layer's outlier eigenspace evolving over the course of training, and exhibiting rank deficiency when the SGD converges to sub-optimal classifiers. This establishes some of the rich predictions that have arisen from extensive numerical studies in the last decade about the spectra of Hessian and information matrices over the course of training in overparametrized networks.

Recent low-rank training methods, such as GaLore, have significantly reduced the memory required to optimize large language models (LLMs). However, these methods often suffer from time-consuming low-rank projection estimations. In particular, the singular value decomposition (SVD) in GaLore can consume more than 80\% of the total training time. To address this issue, we propose GaLore+, which uses cross-head low-rank projection to reduce the substantial time consumption in estimating low-rank projections for multi-head attention. In addition, we employ randomized subspace iteration to achieve fast SVD. To further enhance performance, we propose sparsely coded residuals to reduce the errors caused by low-rank approximation on the first- and second-order moments of the optimizers and weight updates. We evaluate GaLore+ on arithmetic reasoning and natural language generation datasets. Our experiments demonstrate that GaLore+ delivers superior performance while achieving approximately 4times fine-tuning speed compared to vanilla GaLore.

This paper explores how theory can guide and enhance practical algorithms, using Low-Rank Adaptation (LoRA, Hu et al. 2022) in large language models as a case study. We rigorously prove that, under gradient descent, LoRA adapters align with specific singular subspaces of the one-step full fine-tuning gradient. This result suggests that, by properly initializing the adapters using the one-step full gradient, subspace alignment can be achieved immediately and applicable to both linear and nonlinear models. Building on our theory, we propose a theory-driven algorithm, LoRA-One, where the linear convergence (as well as generalization) is built and incorporating preconditioners theoretically helps mitigate the effects of ill-conditioning. Besides, our theory reveals connections between LoRA-One and other gradient-alignment-based methods, helping to clarify misconceptions in the design of such algorithms. LoRA-One achieves significant empirical improvements over LoRA and its variants across benchmarks in natural language understanding, mathematical reasoning, and code generation. Code is available at: https://github.com/YuanheZ/LoRA-One.

Reconstructing 4D spatial intelligence from visual observations has long been a central yet challenging task in computer vision, with broad real-world applications. These range from entertainment domains like movies, where the focus is often on reconstructing fundamental visual elements, to embodied AI, which emphasizes interaction modeling and physical realism. Fueled by rapid advances in 3D representations and deep learning architectures, the field has evolved quickly, outpacing the scope of previous surveys. Additionally, existing surveys rarely offer a comprehensive analysis of the hierarchical structure of 4D scene reconstruction. To address this gap, we present a new perspective that organizes existing methods into five progressive levels of 4D spatial intelligence: (1) Level 1 -- reconstruction of low-level 3D attributes (e.g., depth, pose, and point maps); (2) Level 2 -- reconstruction of 3D scene components (e.g., objects, humans, structures); (3) Level 3 -- reconstruction of 4D dynamic scenes; (4) Level 4 -- modeling of interactions among scene components; and (5) Level 5 -- incorporation of physical laws and constraints. We conclude the survey by discussing the key challenges at each level and highlighting promising directions for advancing toward even richer levels of 4D spatial intelligence. To track ongoing developments, we maintain an up-to-date project page: https://github.com/yukangcao/Awesome-4D-Spatial-Intelligence.

We propose transferability from Large Geometric Vicinity (LGV), a new technique to increase the transferability of black-box adversarial attacks. LGV starts from a pretrained surrogate model and collects multiple weight sets from a few additional training epochs with a constant and high learning rate. LGV exploits two geometric properties that we relate to transferability. First, models that belong to a wider weight optimum are better surrogates. Second, we identify a subspace able to generate an effective surrogate ensemble among this wider optimum. Through extensive experiments, we show that LGV alone outperforms all (combinations of) four established test-time transformations by 1.8 to 59.9 percentage points. Our findings shed new light on the importance of the geometry of the weight space to explain the transferability of adversarial examples.

In this work, we present a variety of novel information-theoretic generalization bounds for learning algorithms, from the supersample setting of Steinke & Zakynthinou (2020)-the setting of the "conditional mutual information" framework. Our development exploits projecting the loss pair (obtained from a training instance and a testing instance) down to a single number and correlating loss values with a Rademacher sequence (and its shifted variants). The presented bounds include square-root bounds, fast-rate bounds, including those based on variance and sharpness, and bounds for interpolating algorithms etc. We show theoretically or empirically that these bounds are tighter than all information-theoretic bounds known to date on the same supersample setting.

We propose 3D Super Resolution (3DSR), a novel 3D Gaussian-splatting-based super-resolution framework that leverages off-the-shelf diffusion-based 2D super-resolution models. 3DSR encourages 3D consistency across views via the use of an explicit 3D Gaussian-splatting-based scene representation. This makes the proposed 3DSR different from prior work, such as image upsampling or the use of video super-resolution, which either don't consider 3D consistency or aim to incorporate 3D consistency implicitly. Notably, our method enhances visual quality without additional fine-tuning, ensuring spatial coherence within the reconstructed scene. We evaluate 3DSR on MipNeRF360 and LLFF data, demonstrating that it produces high-resolution results that are visually compelling, while maintaining structural consistency in 3D reconstructions. Code will be released.

Visual retrieval systems face significant challenges when updating models with improved representations due to misalignment between the old and new representations. The costly and resource-intensive backfilling process involves recalculating feature vectors for images in the gallery set whenever a new model is introduced. To address this, prior research has explored backward-compatible training methods that enable direct comparisons between new and old representations without backfilling. Despite these advancements, achieving a balance between backward compatibility and the performance of independently trained models remains an open problem. In this paper, we address it by expanding the representation space with additional dimensions and learning an orthogonal transformation to achieve compatibility with old models and, at the same time, integrate new information. This transformation preserves the original feature space's geometry, ensuring that our model aligns with previous versions while also learning new data. Our Orthogonal Compatible Aligned (OCA) approach eliminates the need for re-indexing during model updates and ensures that features can be compared directly across different model updates without additional mapping functions. Experimental results on CIFAR-100 and ImageNet-1k demonstrate that our method not only maintains compatibility with previous models but also achieves state-of-the-art accuracy, outperforming several existing methods.

Pre-trained language models, trained on large-scale corpora, demonstrate strong generalizability across various NLP tasks. Fine-tuning these models for specific tasks typically involves updating all parameters, which is resource-intensive. Parameter-efficient fine-tuning (PEFT) methods, such as the popular LoRA family, introduce low-rank matrices to learn only a few parameters efficiently. However, during inference, the product of these matrices updates all pre-trained parameters, complicating tasks like knowledge editing that require selective updates. We propose a novel PEFT method, which conducts row and column-wise sparse low-rank adaptation (RoseLoRA), to address this challenge. RoseLoRA identifies and updates only the most important parameters for a specific task, maintaining efficiency while preserving other model knowledge. By adding a sparsity constraint on the product of low-rank matrices and converting it to row and column-wise sparsity, we ensure efficient and precise model updates. Our theoretical analysis guarantees the lower bound of the sparsity with respective to the matrix product. Extensive experiments on five benchmarks across twenty datasets demonstrate that RoseLoRA outperforms baselines in both general fine-tuning and knowledge editing tasks.

In this work, we study how well the learned weights of a neural network utilize the space available to them. This notion is related to capacity, but additionally incorporates the interaction of the network architecture with the dataset. Most learned weights appear to be full rank, and are therefore not amenable to low rank decomposition. This deceptively implies that the weights are utilizing the entire space available to them. We propose a simple data-driven transformation that projects the weights onto the subspace where the data and the weight interact. This preserves the functional mapping of the layer and reveals its low rank structure. In our findings, we conclude that most models utilize a fraction of the available space. For instance, for ViTB-16 and ViTL-16 trained on ImageNet, the mean layer utilization is 35% and 20% respectively. Our transformation results in reducing the parameters to 50% and 25% respectively, while resulting in less than 0.2% accuracy drop after fine-tuning. We also show that self-supervised pre-training drives this utilization up to 70%, justifying its suitability for downstream tasks.

This work explores the in-context learning capabilities of State Space Models (SSMs) and presents, to the best of our knowledge, the first theoretical explanation of a possible underlying mechanism. We introduce a novel weight construction for SSMs, enabling them to predict the next state of any dynamical system after observing previous states without parameter fine-tuning. This is accomplished by extending the HiPPO framework to demonstrate that continuous SSMs can approximate the derivative of any input signal. Specifically, we find an explicit weight construction for continuous SSMs and provide an asymptotic error bound on the derivative approximation. The discretization of this continuous SSM subsequently yields a discrete SSM that predicts the next state. Finally, we demonstrate the effectiveness of our parameterization empirically. This work should be an initial step toward understanding how sequence models based on SSMs learn in context.

Federated learning algorithms, such as FedAvg, are negatively affected by data heterogeneity and partial client participation. To mitigate the latter problem, global variance reduction methods, like FedVARP, leverage stale model updates for non-participating clients. These methods are effective under homogeneous client participation. Yet, this paper shows that, when some clients participate much less than others, aggregating updates with different levels of staleness can detrimentally affect the training process. Motivated by this observation, we introduce FedStale, a novel algorithm that updates the global model in each round through a convex combination of "fresh" updates from participating clients and "stale" updates from non-participating ones. By adjusting the weight in the convex combination, FedStale interpolates between FedAvg, which only uses fresh updates, and FedVARP, which treats fresh and stale updates equally. Our analysis of FedStale convergence yields the following novel findings: i) it integrates and extends previous FedAvg and FedVARP analyses to heterogeneous client participation; ii) it underscores how the least participating client influences convergence error; iii) it provides practical guidelines to best exploit stale updates, showing that their usefulness diminishes as data heterogeneity decreases and participation heterogeneity increases. Extensive experiments featuring diverse levels of client data and participation heterogeneity not only confirm these findings but also show that FedStale outperforms both FedAvg and FedVARP in many settings.

Maximizing monotone submodular functions under a matroid constraint is a classic algorithmic problem with multiple applications in data mining and machine learning. We study this classic problem in the fully dynamic setting, where elements can be both inserted and deleted in real-time. Our main result is a randomized algorithm that maintains an efficient data structure with an O(k^2) amortized update time (in the number of additions and deletions) and yields a 4-approximate solution, where k is the rank of the matroid.

In the training of large language models (LLMs), updating parameters more efficiently and stably has always been an important challenge. To achieve efficient parameter updates, existing methods usually achieve performance comparable to full parameter updates through methods such as low-dimensional decomposition or layer-wise selective updates. In this work, we propose AlphaAdam, an optimization framework for LLM from the perspective of intra-layer parameter updates. By decoupling parameter updates and dynamically adjusting their strength, AlphaAdam accelerates convergence and improves training stability. We construct parameter masks based on the consistency of historical momentum and gradient direction and combine them with an adaptive mask strength strategy to ensure efficient optimization and theoretical convergence guarantees, which is also applicable to most momentum-based optimizers. Extensive experiments show that AlphaAdam outperforms state-of-the-art methods such as AdamW in terms of convergence speed and computational efficiency across tasks, including GPT-2 pre-trained and fine-tuned RoBERTa and Llama-7B. Our AlphaAdam implements an optimizer enhancement framework for LLMs through intra-layer asynchronous masked adaptive updates. Our code is available in this https://github.com/MaeChd/AlphaAdam.

Sequential learning with Gaussian processes (GPs) is challenging when access to past data is limited, for example, in continual and active learning. In such cases, errors can accumulate over time due to inaccuracies in the posterior, hyperparameters, and inducing points, making accurate learning challenging. Here, we present a method to keep all such errors in check using the recently proposed dual sparse variational GP. Our method enables accurate inference for generic likelihoods and improves learning by actively building and updating a memory of past data. We demonstrate its effectiveness in several applications involving Bayesian optimization, active learning, and continual learning.