Daily Papers

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.

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.

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.

Neural network models of early sensory processing typically reduce the dimensionality of streaming input data. Such networks learn the principal subspace, in the sense of principal component analysis (PCA), by adjusting synaptic weights according to activity-dependent learning rules. When derived from a principled cost function these rules are nonlocal and hence biologically implausible. At the same time, biologically plausible local rules have been postulated rather than derived from a principled cost function. Here, to bridge this gap, we derive a biologically plausible network for subspace learning on streaming data by minimizing a principled cost function. In a departure from previous work, where cost was quantified by the representation, or reconstruction, error, we adopt a multidimensional scaling (MDS) cost function for streaming data. The resulting algorithm relies only on biologically plausible Hebbian and anti-Hebbian local learning rules. In a stochastic setting, synaptic weights converge to a stationary state which projects the input data onto the principal subspace. If the data are generated by a nonstationary distribution, the network can track the principal subspace. Thus, our result makes a step towards an algorithmic theory of neural computation.

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).

Vision-language models (VLMs), despite their extraordinary zero-shot capabilities, are vulnerable to distribution shifts. Test-time adaptation (TTA) emerges as a predominant strategy to adapt VLMs to unlabeled test data on the fly. However, existing TTA methods heavily rely on zero-shot predictions as pseudo-labels for self-training, which can be unreliable under distribution shifts and misguide adaptation due to two fundamental limitations. First (Modality Gap), distribution shifts induce gaps between visual and textual modalities, making cross-modal relations inaccurate. Second (Visual Nuisance), visual embeddings encode rich but task-irrelevant noise that often overwhelms task-specific semantics under distribution shifts. To address these limitations, we propose SubTTA, which aligns the semantic subspaces of both modalities to enhance zero-shot predictions to better guide the TTA process. To bridge the modality gap, SubTTA extracts the principal subspaces of both modalities and aligns the visual manifold to the textual semantic anchor by minimizing their chordal distance. To eliminate visual nuisance, SubTTA projects the aligned visual features onto the task-specific textual subspace, which filters out task-irrelevant noise by constraining visual embeddings within the valid semantic span, and standard TTA is further performed on the purified space to refine the decision boundaries. Extensive experiments on various benchmarks and VLM architectures demonstrate the effectiveness of SubTTA, yielding an average improvement of 2.24% over state-of-the-art TTA methods.

Mixture-of-Experts (MoE) scales model capacity efficiently by selectively routing inputs to a specialized subset of experts. However, input-expert specialization, the core motivation of MoE, critically depends on whether the router is actually aware of input structure. In practice, MoE routing is typically implemented as a shallow linear projection with limited awareness of input representation, which often leads to unstable routing. We propose STAR, a Structure Aware Routing that rethinks MoE routing as a subspace learning problem by augmenting standard learnable routing with an evolving principal subspace that tracks dominant input structure via Generalized Hebbian Algorithm (GHA). By aligning routing decisions directly with input structure, STAR enables stable expert specialization. We evaluate STAR on controlled synthetic setup and large-scale language and vision tasks, where it consistently improves routing quality and downstream performance over strong MoE baselines. Moreover, optional test-time subspace updates further enhance routing robustness and generalization under input distribution shifts.

Video world models achieve long-range temporal consistency by storing KV cache during generation, but the growing cache makes KV cache memory a major deployment bottleneck, which motivates low-bit quantization study for efficiency. Existing 2-bit KV cache quantization methods can achieve nearly lossless performance on VBench, however, when applied to video world models, we find they still cause severe temporal flickering and visual degradation. Meanwhile, deeper investigates show that Key quantization produces smaller reconstruction errors than Value, but surprisingly leads to larger output degradation. We trace this discrepancy to attention in video world models: Key perturbations can change the attention logits, and shift the temporal-spatial tokens selected by Queries. These observations motivate us to preserve attention logits and temporal-spatial token selection during KV cache quantization. To address this issue, we present QuantWM, a training-free 2-bit KV cache quantization framework for video world models. QuantWM introduces two complementary techniques to mitigate the attention shifts. Firstly, quantization-sensitivity-aware clustering (QSAC) jointly considers historical Query sensitivity and residual ranges to select INT2-friendly Key centroids, which reduces quantization errors in channels that are more critical to attention. In addition, principal-subspace attention compensation (PSAC) restores the remaining Key errors along the dominant Query subspace using low-rank projections, which provides a direct and efficient correction to stabilize attention logits. Experiments on LingBot-World-v2, HY-World 1.5, Matrix-Game-2, Longcat-Video and Causal-Forcing demonstrate that QuantWM significantly improves visual quality and temporal consistency, while outperforming existing methods across benchmarks with up to 6.20 KV cache memory compression and limited additional overhead.

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.

Neuromorphic computing with spiking neural networks is promising for energy-efficient artificial intelligence (AI) applications. However, different from humans who continually learn different tasks in a lifetime, neural network models suffer from catastrophic forgetting. How could neuronal operations solve this problem is an important question for AI and neuroscience. Many previous studies draw inspiration from observed neuroscience phenomena and propose episodic replay or synaptic metaplasticity, but they are not guaranteed to explicitly preserve knowledge for neuron populations. Other works focus on machine learning methods with more mathematical grounding, e.g., orthogonal projection on high dimensional spaces, but there is no neural correspondence for neuromorphic computing. In this work, we develop a new method with neuronal operations based on lateral connections and Hebbian learning, which can protect knowledge by projecting activity traces of neurons into an orthogonal subspace so that synaptic weight update will not interfere with old tasks. We show that Hebbian and anti-Hebbian learning on recurrent lateral connections can effectively extract the principal subspace of neural activities and enable orthogonal projection. This provides new insights into how neural circuits and Hebbian learning can help continual learning, and also how the concept of orthogonal projection can be realized in neuronal systems. Our method is also flexible to utilize arbitrary training methods based on presynaptic activities/traces. Experiments show that our method consistently solves forgetting for spiking neural networks with nearly zero forgetting under various supervised training methods with different error propagation approaches, and outperforms previous approaches under various settings. Our method can pave a solid path for building continual neuromorphic computing systems.

Continual learning for pre-trained vision-language models requires balancing three competing objectives: retaining pre-trained knowledge, preserving knowledge from a sequence of learned tasks, and maintaining the plasticity to acquire new knowledge. This paper presents a simple but effective approach called KeepLoRA to effectively balance these objectives. We first analyze the knowledge retention mechanism within the model parameter space and find that general knowledge is mainly encoded in the principal subspace, while task-specific knowledge is encoded in the residual subspace. Motivated by this finding, KeepLoRA learns new tasks by restricting LoRA parameter updates in the residual subspace to prevent interfering with previously learned capabilities. Specifically, we infuse knowledge for a new task by projecting its gradient onto a subspace orthogonal to both the principal subspace of pre-trained model and the dominant directions of previous task features. Our theoretical and empirical analyses confirm that KeepLoRA balances the three objectives and achieves state-of-the-art performance. The implementation code is available at https://github.com/MaolinLuo/KeepLoRA.

Reinforcement learning (RL) has become a key paradigm for enhancing the reasoning of large language models, yet the high dimensionality of parameter updates makes its training dynamics hard to analyze. We study reinforcement learning with verifiable rewards (RLVR) and use vector steering to identify a low-dimensional effective manifold in activation space associated with RL-induced gains. We uncover two geometric properties. (1) Effective Manifold Capacity: the capacity needed to reproduce RL gains can be very small but is not infinitely compressible; at extremely low capacity, intervention dimensionality and input-dependent expressiveness become key constraints, and this requirement varies with injection depth. (2) Control Manifold Separation: effective control directions lie mainly in the low-variance complement of the activation principal subspace. Within a task and base model, the learned geometry stays largely consistent across training configurations, and across tasks geometric alignment correlates with capability transfer. Experiments on 5 LLMs and 6 verifiable-reward tasks support these findings. We then propose Alpha-Stabler, a plug-and-play framework with a Predictor that monitors principal-subspace intrusion for early collapse warnings, and a Controller that removes the principal-subspace component of activation gradients during backpropagation while preserving the orthogonal complement. Alpha-Stabler stabilizes training for 2,000 steps and consistently improves RL gains, offering practical insights for robust post-training. Code: https://github.com/caiyuchen-ustc/On_Policy_Vector_Training

Multimodal LLMs are typically trained by first aligning a vision encoder with a language model via projector training on large image-caption datasets, followed by joint training of the full multimodal model. Despite its widespread use, the role of the projector training stage remains poorly understood. We find that (i) using only 10 to 20% of the data for training the projector already recovers most downstream performance gains, and (ii) projectors trained on different datasets can be linearly interpolated while largely preserving performance. These results suggest that projector training mainly performs a coarse alignment, which many different projectors achieve equally well. Motivated by these findings, we propose PORTAL, a training-free projector initialization that requires neither paired image-caption data nor gradient-based optimization, but only per-modality summary statistics. PORTAL computes an optimal-transport-based map between Gaussian approximations of the vision and language feature distributions on a shared principal subspace, so that the projector training stage can be skipped entirely. Without any paired data, PORTAL matches the standard pipeline trained on 558k image-caption pairs across six LLM backbones, two vision encoders, and 16 benchmarks. It outperforms that pipeline in 3 of 7 (vision encoder, LLM) settings and reaches an average 98% of its score across the other four.

Reinforcement Learning with Verifiable Rewards (RLVR) reliably improves the reasoning performance of large language models, yet it appears to modify only a small fraction of parameters. We revisit this paradox and show that sparsity is a surface artifact of a model-conditioned optimization bias: for a fixed pretrained model, updates consistently localize to preferred parameter regions, highly consistent across runs and largely invariant to datasets and RL recipes. We mechanistically explain these dynamics with a Three-Gate Theory: Gate I (KL Anchor) imposes a KL-constrained update; Gate II (Model Geometry) steers the step off principal directions into low-curvature, spectrum-preserving subspaces; and Gate III (Precision) hides micro-updates in non-preferred regions, making the off-principal bias appear as sparsity. We then validate this theory and, for the first time, provide a parameter-level characterization of RLVR's learning dynamics: RLVR learns off principal directions in weight space, achieving gains via minimal spectral drift, reduced principal-subspace rotation, and off-principal update alignment. In contrast, SFT targets principal weights, distorts the spectrum, and even lags RLVR. Together, these results provide the first parameter-space account of RLVR's training dynamics, revealing clear regularities in how parameters evolve. Crucially, we show that RL operates in a distinct optimization regime from SFT, so directly adapting SFT-era parameter-efficient fine-tuning (PEFT) methods can be flawed, as evidenced by our case studies on advanced sparse fine-tuning and LoRA variants. We hope this work charts a path toward a white-box understanding of RLVR and the design of geometry-aware, RLVR-native learning algorithms, rather than repurposed SFT-era heuristics.

Precise emotion control in audio-driven talking heads remains a challenge due to the reliance on implicit emotion regulation in existing systems, which often leads to indirect and insufficient control. Additionally, training with explicit emotion-related losses across the entire motion space poses significant difficulties due to the inherent trade-off between accurate lip synchronization and fine-grained emotion control. In this paper, we reveal a key finding: although emotional cues are distributed throughout the motion space, concentrating discriminative supervision on less-principal components achieves a better emotion-lip synchronization balance, as principal components mainly encode high-energy articulation and pose variations. Building on this insight, we propose Xemo-Talker, which first learns a neutral speech-to-motion mapping for stable articulation and lip synchronization, and then introduces a lightweight emotion branch guided by less-principal subspace supervision. To enhance emotion control, we design a Tri-Loss consisting of inter-class separation, intra-class compactness, and less-principal contrastive learning. Given an audio input, a reference image, and an emotion label, Xemo-Talker achieves state-of-the-art emotion classification accuracy while maintaining competitive lip synchronization and high inference efficiency, with performance approaching that measured on real videos.The source code is publicly available at https://github.com/chaolongy/Xemo-Talker.

Large language models generalize well to individual tasks but lack an inherent mechanism for learning them sequentially, leading to catastrophic forgetting. To mitigate this, LoRA-based continual learning methods allocate a separate low-rank adapter per task, yet existing approaches either require task identity at inference or sum all adapters indiscriminately, letting irrelevant branches distort the output. Recent gating-based solutions route inputs to the correct adapter but introduce trainable parameters that themselves need protection against forgetting. In this work, we observe that pooled token embeddings from a frozen LLM embedding layer already separate task distributions throughout the learning sequence. A Gaussian mixture model fitted on these embeddings, without any gradient-based training, is sufficient for task-agnostic adapter selection at test time. This eliminates the need for a learned gating module. On the adapter side, constraining each task's parameters to the principal subspace of the pretrained weights via SVD yields a compact latent-space parameterization. Within this subspace, orthogonal regularization directly controls inter-task interference. The resulting system, Latent-LoRA, is replay-free, requires no trainable routing component, and uses substantially fewer parameters per task. Experiments across five model scales and two established continual learning benchmarks show state-of-the-art performance with near-zero forgetting.

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.

On-policy distillation (OPD) has recently become a prominent post-training recipe as it combines two desirable ingredients: on-policy student trajectories and dense teacher supervision, yet how this hybrid changes a model's parameters remains unclear. Across several language and vision-language model pairs and use cases, our analysis yields two main findings. On sparsity, OPD-style updates are small and coordinate-sparse. They are distributed across layers and are usually FFN-heavy. This sparse structure is operationally useful: training only the discovered subnetwork recovers nearly the same performance as full OPD. However, the sparsity-inducing SGD optimizer underperforms AdamW in our optimizer ablation, likely because dense teacher supervision preserves heterogeneous coordinate-wise gradient scales where AdamW's adaptive scaling remains useful. On geometry, the updates are numerically full-rank but spectrally concentrated; they lie mostly away from the principal singular subspaces of the source weights and fall disproportionately on coordinates where the source weights are close to zero. These findings suggest that dense teacher supervision does not turn OPD into ordinary dense parameter rewriting; instead, OPD retains important geometric signatures of on-policy post-training.

On-policy distillation (OPD) is increasingly used to improve large language model reasoning, but its training dynamics remain poorly understood. We characterize the trajectory of OPD updates in parameter space and compare it with supervised fine-tuning (SFT) and reinforcement learning with verifiable rewards (RLVR). A suite of parameter-space diagnostics consistently places OPD in a relaxed off-principal regime: compared with SFT, its updates affect fewer weights and avoid principal directions more strongly, while compared with RLVR, they remain less tightly constrained. Beyond this static localization, OPD exhibits subspace locking: its cumulative updates rapidly enter a narrow low-dimensional channel. Constraining training to the update subspace formed early in training preserves OPD performance but substantially degrades SFT, indicating that the locked subspace is functionally sufficient for OPD. Control experiments further show that sparsifying the update tokens and shifting rollout generation off-policy preserve the rank dynamics, whereas mixing the OPD objective with RLVR changes them. Overall, these results suggest that OPD is not merely an intermediate point between SFT and RLVR, but induces its own update geometry in parameter space.

Large Language Models (LLMs) have shown improved generation performance through retrieval-augmented generation (RAG) following the retriever-reader paradigm, which supplements model inputs with externally retrieved knowledge. However, prior work often evaluates RAG holistically, assessing the retriever and reader jointly, making it difficult to isolate the true contribution of retrieval, particularly given the prompt sensitivity of LLMs used as readers. We introduce Spectrum Projection Score (SPS), a lightweight, supervision-free metric that allows the reader to gauge the semantic alignment of a retrieved summary with its hidden representation by comparing the area formed by generated tokens from the summary, and the principal directions of subspace in the reader and to measure the relevance. Building on SPS we present xCompress, an inference time controller framework that dynamically samples, ranks, and compresses retrieval summary candidates. Extensive experiments on five QA benchmarks with four open source LLMs show that SPS not only enhances performance across a range of tasks but also provides a principled perspective on the interaction between retrieval and generation.

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.

Model merging aims to integrate multiple task-specific fine-tuned models derived from a shared pre-trained checkpoint into a single multi-task model without additional training. Despite extensive research, task interference remains a major obstacle that often undermines the performance of merged models. In this paper, we propose ESM (Essential Subspace Merging) , a robust framework for effective model merging. We begin by performing Principal Component Analysis (PCA) on feature shifts induced by parameter updates. The resulting principal directions span an essential subspace that dominantly influences feature representations. Each task's parameter update matrix is projected onto its respective essential subspace for low-rank decomposition before merging. This methodology mitigates inter-task interference while preserving core task-specific functionality. Furthermore, we introduce a multi-level polarized scaling strategy that amplifies parameters containing critical knowledge and suppresses redundant ones, preventing essential knowledge from being overwhelmed during fusion. Extensive experiments across multiple task sets and model scales demonstrate that our method achieves state-of-the-art performance in multi-task model merging.

Detecting visual anomalies in industrial inspection often requires training with only a few normal images per category. Recent few-shot methods achieve strong results employing foundation-model features, but typically rely on memory banks, auxiliary datasets, or multi-modal tuning of vision-language models. We therefore question whether such complexity is necessary given the feature representations of vision foundation models. To answer this question, we introduce SubspaceAD, a training-free method, that operates in two simple stages. First, patch-level features are extracted from a small set of normal images by a frozen DINOv2 backbone. Second, a Principal Component Analysis (PCA) model is fit to these features to estimate the low-dimensional subspace of normal variations. At inference, anomalies are detected via the reconstruction residual with respect to this subspace, producing interpretable and statistically grounded anomaly scores. Despite its simplicity, SubspaceAD achieves state-of-the-art performance across one-shot and few-shot settings without training, prompt tuning, or memory banks. In the one-shot anomaly detection setting, SubspaceAD achieves image-level and pixel-level AUROC of 97.1% and 97.5% on the MVTec-AD dataset, and 93.4% and 98.2% on the VisA dataset, respectively, surpassing prior state-of-the-art results. Code and demo are available at https://github.com/CLendering/SubspaceAD.

Parameter-efficient fine-tuning (PEFT) methods, such as LoRA, enable scalable adaptation of foundation models by injecting low-rank adapters. However, their communication and storage costs remain a major bottleneck in resource-constrained settings. We propose SOLAR (Subspace-Oriented Latent Adapter Reparameterization), a post-training compression framework that substantially reduces the communication cost (i.e., the number of parameters to transmit or store) of PEFT adapters. SOLAR expresses each PEFT update as a linear combination of basis vectors formed from the foundation model's singular vectors with controlled random perturbations. By exploiting the subspace similarity (the alignment of principal directions) between the foundation model and task-specific fine-tuned updates, SOLAR decouples the adapter size from PEFT structure and ensures compact yet expressive representations. It is model-agnostic and compatible with existing PEFT methods, including LoRA, AdaLoRA, and other adapter modules. We theoretically establish a bound on the reconstruction error. Experiments on language and vision tasks using LLaMA, GPT, and ViT models demonstrate that SOLAR preserves task performance while significantly reducing model representation sizes, offering an effective and communication-efficient solution for deployment in distributed systems and edge devices.

Model merging aims to enable multi-task learning by integrating the capabilities of multiple models fine-tuned from the same pre-trained checkpoint into a single model. Its core challenge is inter-task interference among task-specific parameter updates. In this paper, we analyze the output shifts induced by task updates and observe that their energy is concentrated in a small number of principal directions. We call the subspace spanned by these directions the essential subspace. In contrast, most remaining directions carry little task-relevant energy, but their accumulation across multiple task updates can cause severe interference during merging. Motivated by this observation, we propose Essential Subspace Decomposition (ESD), which decomposes each task update according to the principal components of its activation shift. Based on ESD, we introduce Essential Subspace Merging (ESM), a training-free static merging method that orthogonalizes and fuses essential components into one compact multi-task model. We further extend ESM to ESM++, a training-free dynamic merging method that decomposes task-specific residuals into low-rank experts and selects the most relevant expert through prototype-based routing during forward inference. Extensive experiments across multiple task sets and model scales demonstrate that ESM and ESM++ effectively preserves task knowledge while reducing inter-task interference.

Domain Generalization (DG) aims to learn representations that remain robust under out-of-distribution (OOD) shifts and generalize effectively to unseen target domains. While recent invariant learning strategies and architectural advances have achieved strong performance, explicitly discovering a structured domain-invariant subspace through second-order statistics remains underexplored. In this work, we propose CPCANet, a novel framework grounded in Common Principal Component Analysis (CPCA), which unrolls the iterative Flury-Gautschi (FG) algorithm into fully differentiable neural layers. This approach integrates the statistical properties of CPCA into an end-to-end trainable framework, enforcing the discovery of a shared subspace across diverse domains while preserving interpretability. Experiments on four standard DG benchmarks demonstrate that CPCANet achieves state-of-the-art (SOTA) performance in zero-shot transfer. Moreover, CPCANet is architecture-agnostic and requires no dataset-specific tuning, providing a simple and efficient approach to learning robust representations under distribution shift. Code is available at https://github.com/wish44165/CPCANet.

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.

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.

In the era of Internet of Things (IoT), network-wide anomaly detection is a crucial part of monitoring IoT networks due to the inherent security vulnerabilities of most IoT devices. Principal Components Analysis (PCA) has been proposed to separate network traffics into two disjoint subspaces corresponding to normal and malicious behaviors for anomaly detection. However, the privacy concerns and limitations of devices' computing resources compromise the practical effectiveness of PCA. We propose a federated PCA-based Grassmannian optimization framework that coordinates IoT devices to aggregate a joint profile of normal network behaviors for anomaly detection. First, we introduce a privacy-preserving federated PCA framework to simultaneously capture the profile of various IoT devices' traffic. Then, we investigate the alternating direction method of multipliers gradient-based learning on the Grassmann manifold to guarantee fast training and the absence of detecting latency using limited computational resources. Empirical results on the NSL-KDD dataset demonstrate that our method outperforms baseline approaches. Finally, we show that the Grassmann manifold algorithm is highly adapted for IoT anomaly detection, which permits drastically reducing the analysis time of the system. To the best of our knowledge, this is the first federated PCA algorithm for anomaly detection meeting the requirements of IoT networks.

Text-to-image diffusion models, such as Stable Diffusion, have demonstrated remarkable capabilities in generating high-quality and diverse images from natural language prompts. However, recent studies reveal that these models often replicate and amplify societal biases, particularly along demographic attributes like gender and race. In this paper, we introduce FairImagen (https://github.com/fuzihaofzh/FairImagen), a post-hoc debiasing framework that operates on prompt embeddings to mitigate such biases without retraining or modifying the underlying diffusion model. Our method integrates Fair Principal Component Analysis to project CLIP-based input embeddings into a subspace that minimizes group-specific information while preserving semantic content. We further enhance debiasing effectiveness through empirical noise injection and propose a unified cross-demographic projection method that enables simultaneous debiasing across multiple demographic attributes. Extensive experiments across gender, race, and intersectional settings demonstrate that FairImagen significantly improves fairness with a moderate trade-off in image quality and prompt fidelity. Our framework outperforms existing post-hoc methods and offers a simple, scalable, and model-agnostic solution for equitable text-to-image generation.

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.

Existing AI-generated image (AIGI) detection methods often suffer from limited generalization performance. In this paper, we identify a crucial yet previously overlooked asymmetry phenomenon in AIGI detection: during training, models tend to quickly overfit to specific fake patterns in the training set, while other information is not adequately captured, leading to poor generalization when faced with new fake methods. A key insight is to incorporate the rich semantic knowledge embedded within large-scale vision foundation models (VFMs) to expand the previous discriminative space (based on forgery patterns only), such that the discrimination is decided by both forgery and semantic cues, thereby reducing the overfitting to specific forgery patterns. A straightforward solution is to fully fine-tune VFMs, but it risks distorting the well-learned semantic knowledge, pushing the model back toward overfitting. To this end, we design a novel approach called Effort: Efficient orthogonal modeling for generalizable AIGI detection. Specifically, we employ Singular Value Decomposition (SVD) to construct the orthogonal semantic and forgery subspaces. By freezing the principal components and adapting the residual components (sim0.19M parameters), we preserve the original semantic subspace and use its orthogonal subspace for learning forgeries. Extensive experiments on AIGI detection benchmarks demonstrate the superior effectiveness of our approach.

Efficient finetuning of large language models (LLMs) aims to adapt the LLMs with reduced computational and memory cost. Previous LoRA-based approaches initialize the low-rank matrices with Gaussian distribution and zero values while keeping the original weight matrices frozen. However, the trainable model parameters optimized in an unguided subspace might interfere with the well-learned subspace of the pretrained weight matrices. In this paper, we propose MiLoRA, a simple yet effective LLM finetuning approach that only updates the minor singular components of the weight matrix while keeping the principal singular components frozen. It is observed that the minor matrix corresponds to the noisy or long-tail information, while the principal matrix contains important knowledge. The MiLoRA initializes the low-rank matrices within a subspace that is orthogonal to the principal matrix, thus the pretrained knowledge is expected to be well preserved. During finetuning, MiLoRA makes the most use of the less-optimized subspace for learning the labeled dataset. Extensive experiments on commonsense reasoning, math reasoning, instruction following and visual instruction following benchmarks present the superior performance of our method.

Graph neural networks aim to learn representations for graph-structured data and show impressive performance, particularly in node classification. Recently, many methods have studied the representations of GNNs from the perspective of optimization goals and spectral graph theory. However, the feature space that dominates representation learning has not been systematically studied in graph neural networks. In this paper, we propose to fill this gap by analyzing the feature space of both spatial and spectral models. We decompose graph neural networks into determined feature spaces and trainable weights, providing the convenience of studying the feature space explicitly using matrix space analysis. In particular, we theoretically find that the feature space tends to be linearly correlated due to repeated aggregations. Motivated by these findings, we propose 1) feature subspaces flattening and 2) structural principal components to expand the feature space. Extensive experiments verify the effectiveness of our proposed more comprehensive feature space, with comparable inference time to the baseline, and demonstrate its efficient convergence capability.

Vision-Language Models (VLMs) excel at zero-shot inference but often degrade under test-time domain shifts. For this reason, episodic test-time adaptation strategies have recently emerged as powerful techniques for adapting VLMs to a single unlabeled image. However, existing adaptation strategies, such as test-time prompt tuning, typically require backpropagating through large encoder weights or altering core model components. In this work, we introduce Spectrum-Aware Test-Time Steering (STS), a lightweight adaptation framework that extracts a spectral subspace from the textual embeddings to define principal semantic directions and learns to steer latent representations in a spectrum-aware manner by adapting a small number of per-sample shift parameters to minimize entropy across augmented views. STS operates entirely at inference in the latent space, without backpropagation through or modification of the frozen encoders. Building on standard evaluation protocols, our comprehensive experiments demonstrate that STS largely surpasses or compares favorably against state-of-the-art test-time adaptation methods, while introducing only a handful of additional parameters and achieving inference speeds up to 8x faster with a 12x smaller memory footprint than conventional test-time prompt tuning. The code is available at https://github.com/kdafnis/STS.

Vision-Language Models (VLMs) increasingly power high-stakes applications, from medical imaging to autonomous systems, yet they routinely hallucinate, confidently describing content not present in the input. We investigate the root causes of these failure modes with a mechanistic analysis focusing on the decoder-based VLMs. We trace these failure modes to a geometric over-alignment: to bridge the modality gap required by attention mechanisms, decoder-based VLMs over-align visual embeddings with the text manifold, injecting a statistical linguistic bias that systematically overshadows fine-grained visual evidence. While prior work either aggressively closes this gap or suppresses hallucinations through expensive black-box decoding strategies, none addresses the underlying geometric cause. We provide the first quantitative characterization of this over-alignment, demonstrating that linguistic bias concentrates in the top principal components of a universal, dataset-agnostic text subspace. Building on this insight, we propose two complementary remedies: a training-free inference strategy and a bias-aware fine-tuning paradigm, both of which explicitly project out this subspace from visual representations. Our methods significantly reduce hallucinations across POPE, CHAIR, and AMBER benchmarks, and improve CLAIR scores on long-form captioning tasks, with the training-free variant adding no computational overhead over the base model.

Cross-entropy pretraining and preference alignment update the same transformer weights, but leave geometrically distinct traces. We characterise this asymmetry with a relative-subspace-fraction probe that tracks how weight deltas align with residual-stream activation subspaces and with the prediction subspace defined by the unembedding. Alignment deltas concentrate in the read pathway (W_Q, W_K), along principal directions of attention-input activations, while remaining near-isotropic in the write pathway (W_O, W_2) relative to the prediction subspace. We explain this pattern through anisotropic gradient accumulation: updates to a matrix W are sums of outer products δ_t a_t^top, and inherit directional structure from whichever side has concentrated covariance. For read-pathway matrices, this side is the input activation a_t, whose covariance is spiked in trained transformers and therefore produces objective-agnostic concentration. For write-pathway matrices, the relevant side is the upstream gradient δ_t, whose anisotropy depends on the loss. Cross-entropy supplies the canonical sharp per-sample signal, inducing write-pathway prediction geometry during pretraining; alignment objectives typically add little further write-side concentration. We support this explanation with a within-checkpoint trajectory, a graded contrastive-objective control, and a closed-form rank-1 intervention with matched direction controls, providing causal evidence for the proposed weight-space geometry.

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.

We study principal component analysis (PCA), where given a dataset in R^d from a distribution, the task is to find a unit vector v that approximately maximizes the variance of the distribution after being projected along v. Despite being a classical task, standard estimators fail drastically if the data contains even a small fraction of outliers, motivating the problem of robust PCA. Recent work has developed computationally-efficient algorithms for robust PCA that either take super-linear time or have sub-optimal error guarantees. Our main contribution is to develop a nearly-linear time algorithm for robust PCA with near-optimal error guarantees. We also develop a single-pass streaming algorithm for robust PCA with memory usage nearly-linear in the dimension.

Principal component analysis (PCA) is a mainstay of modern data analysis - a black box that is widely used but (sometimes) poorly understood. The goal of this paper is to dispel the magic behind this black box. This manuscript focuses on building a solid intuition for how and why principal component analysis works. This manuscript crystallizes this knowledge by deriving from simple intuitions, the mathematics behind PCA. This tutorial does not shy away from explaining the ideas informally, nor does it shy away from the mathematics. The hope is that by addressing both aspects, readers of all levels will be able to gain a better understanding of PCA as well as the when, the how and the why of applying this technique.

Principal component analysis (PCA) is a tool to capture factors that explain variation in data. Across domains, data are now collected across multiple contexts (for example, individuals with different diseases, cells of different types, or words across texts). While the factors explaining variation in data are undoubtedly shared across subsets of contexts, no tools currently exist to systematically recover such factors. We develop multi-context principal component analysis (MCPCA), a theoretical and algorithmic framework that decomposes data into factors shared across subsets of contexts. Applied to gene expression, MCPCA reveals axes of variation shared across subsets of cancer types and an axis whose variability in tumor cells, but not mean, is associated with lung cancer progression. Applied to contextualized word embeddings from language models, MCPCA maps stages of a debate on human nature, revealing a discussion between science and fiction over decades. These axes are not found by combining data across contexts or by restricting to individual contexts. MCPCA is a principled generalization of PCA to address the challenge of understanding factors underlying data across contexts.

Subspace segmentation or subspace learning is a challenging and complicated task in machine learning. This paper builds a primary frame and solid theoretical bases for the minimal subspace segmentation (MSS) of finite samples. Existence and conditional uniqueness of MSS are discussed with conditions generally satisfied in applications. Utilizing weak prior information of MSS, the minimality inspection of segments is further simplified to the prior detection of partitions. The MSS problem is then modeled as a computable optimization problem via self-expressiveness of samples. A closed form of representation matrices is first given for the self-expressiveness, and the connection of diagonal blocks is then addressed. The MSS model uses a rank restriction on the sum of segment ranks. Theoretically, it can retrieve the minimal sample subspaces that could be heavily intersected. The optimization problem is solved via a basic manifold conjugate gradient algorithm, alternative optimization and hybrid optimization, taking into account of solving both the primal MSS problem and its pseudo-dual problem. The MSS model is further modified for handling noisy data, and solved by an ADMM algorithm. The reported experiments show the strong ability of the MSS method on retrieving minimal sample subspaces that are heavily intersected.

A popular way to approximate the Koopman operator's action on a finite-dimensional subspace of functions is via orthogonal projections. The quality of the projected model directly depends on the selected subspace, specifically on how close it is to being invariant under the Koopman operator. The notion of invariance proximity provides a tight upper bound on the worst-case relative prediction error of the finite-dimensional model. However, its direct calculation is computationally challenging. This paper leverages the geometric structure behind the definition of invariance proximity to provide a closed-form expression in terms of Jordan principal angles on general inner product spaces. Unveiling this connection allows us to exploit specific isomorphisms to circumvent the computational challenges associated with spaces of functions and enables the use of existing efficient numerical routines to compute invariance proximity.

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.

The exploration of linear subspaces, particularly scattered subspaces, has garnered considerable attention across diverse mathematical disciplines in recent years, notably within finite geometries and coding theory. Scattered subspaces play a pivotal role in analyzing various geometric structures such as blocking sets, two-intersection sets, complete arcs, caps in affine and projective spaces over finite fields and rank metric codes. This paper introduces a new infinite family of h-subspaces, along with their associated MRD codes. Additionally, it addresses the task of determining the generalized weights of these codes. Notably, we demonstrate that these MRD codes exhibit some larger generalized weights compared to those previously identified.

In language models, what a representation encodes is determined by the geometry of its representation space: distances, not activations, carry meaning. Existing tools characterize the shape of this geometry but do not ask what that shape is organized for. We introduce Subspace PGA, a metric that tests whether a layer's distance structure aligns with the readout subspace of the unembedding matrix W_U more than with random subspaces of equal size. Across seven Pythia models (70M--6.9B) and three cross-family models, intermediate geometry is significantly organized for prediction (peak z = 9--24), but the degree is scale-dependent: small models (d leq 1024) progressively lose it at late layers during training -- even as loss keeps improving -- while large models (d geq 2048) preserve it throughout. We trace this to a capacity trade-off: a few dominant directions migrate away from W_U's readout, masking rather than destroying the predictive structure beneath, and removing them restores alignment. Neither spectral metrics nor loss curves capture this distinction. Scale thus determines not only how well a model predicts, but how its representation geometry is organized to do so.

Dimensionality reduction is critical for deploying dense retrieval systems at scale, yet mainstream post-hoc methods face a fundamental trade-off: principal component analysis (PCA) preserves dominant variance but underutilizes representational capacity, while whitening enforces isotropy at the cost of amplifying noise in the heavy-tailed eigenspectrum of retrieval embeddings. Intermediate spectral scaling methods unify these extremes by reweighting dimensions with a power coefficient γ, but treat γ as a fixed hyperparameter that requires task-specific tuning. We show that the optimal scaling strength γ is not a global constant: it varies systematically with target dimensionality k and is governed by the signal-to-noise ratio (SNR) of the retained subspace. Based on this insight, we propose Spectral Tempering (SpecTemp), a learning-free method that derives an adaptive γ(k) directly from the corpus eigenspectrum using local SNR analysis and knee-point normalization, requiring no labeled data or validation-based search. Extensive experiments demonstrate that Spectral Tempering consistently achieves near-oracle performance relative to grid-searched γ^*(k) while remaining fully learning-free and model-agnostic. Our code is publicly available at https://anonymous.4open.science/r/SpecTemp-0D37.

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.

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.

Identifying low-dimensional latent structures within high-dimensional data has long been a central topic in the machine learning community, driven by the need for data compression, storage, transmission, and deeper data understanding. Traditional methods, such as principal component analysis (PCA) and autoencoders (AE), operate in an unsupervised manner, ignoring label information even when it is available. In this work, we introduce a unified method capable of learning latent spaces in both unsupervised and supervised settings. We formulate the problem as a nonlinear multiple-response regression within an index model context. By applying the generalized Stein's lemma, the latent space can be estimated without knowing the nonlinear link functions. Our method can be viewed as a nonlinear generalization of PCA. Moreover, unlike AE and other neural network methods that operate as "black boxes", our approach not only offers better interpretability but also reduces computational complexity while providing strong theoretical guarantees. Comprehensive numerical experiments and real data analyses demonstrate the superior performance of our method.

We present a parallel data-driven strategy to identify finite-dimensional functional spaces invariant under the Koopman operator associated to an unknown dynamical system. We build on the Symmetric Subspace Decomposition (SSD) algorithm, a centralized method that under mild conditions on data sampling provably finds the maximal Koopman-invariant subspace and all Koopman eigenfunctions in an arbitrary finite-dimensional functional space. A network of processors, each aware of a common dictionary of functions and equipped with a local set of data snapshots, repeatedly interact over a directed communication graph. Each processor receives its neighbors' estimates of the invariant dictionary and refines its estimate by applying SSD with its local data on the intersection of the subspaces spanned by its own dictionary and the neighbors' dictionaries. We identify conditions on the network topology to ensure the algorithm identifies the maximal Koopman-invariant subspace in the span of the original dictionary, characterize its time, computational, and communication complexity, and establish its robustness against communication failures.

Principal Component Analysis (PCA) is a popular tool for dimensionality reduction and feature extraction in data analysis. There is a probabilistic version of PCA, known as Probabilistic PCA (PPCA). However, standard PCA and PPCA are not robust, as they are sensitive to outliers. To alleviate this problem, this paper introduces the Self-Paced Learning mechanism into PPCA, and proposes a novel method called Self-Paced Probabilistic Principal Component Analysis (SP-PPCA). Furthermore, we design the corresponding optimization algorithm based on the alternative search strategy and the expectation-maximization algorithm. SP-PPCA looks for optimal projection vectors and filters out outliers iteratively. Experiments on both synthetic problems and real-world datasets clearly demonstrate that SP-PPCA is able to reduce or eliminate the impact of outliers.

During SGD training, the gradients often align strongly with the dominant subspace spanned by the top-k eigenvectors of the Hessian of the loss. While this seems to naturally imply that loss reduction mainly occurs within this space, prior work has shown that updates within this dominant subspace make no meaningful progress in reducing the loss. In this work, we argue that the dominant subspace is better understood not as the main space for loss reduction, but as a key subspace for explaining the sharpness dynamics of mini-batch SGD. To explain the role of the dominant subspace in reducing top-k sharpness, we show how the averaged gradient over fluctuations in the dominant directions produces a sharpness correction term, and derive a sharpness correction term induced by mini-batch noise in the dominant directions. Experimental results show that adding the derived correction term to GD brings the sharpness evolution of GD closer to that of SGD.

The Distributional Principal Autoencoder (DPA) combines distributionally correct reconstruction with principal-component-like interpretability of the encodings. In this work, we provide exact theoretical guarantees on both fronts. First, we derive a closed-form relation linking each optimal level-set geometry to the data-distribution score. This result explains DPA's empirical ability to disentangle factors of variation of the data, as well as allows the score to be recovered directly from samples. When the data follows the Boltzmann distribution, we demonstrate that this relation yields an approximation of the minimum free-energy path for the Mueller-Brown potential in a single fit. Second, we prove that if the data lies on a manifold that can be approximated by the encoder, latent components beyond the manifold dimension are conditionally independent of the data distribution - carrying no additional information - and thus reveal the intrinsic dimension. Together, these results show that a single model can learn the data distribution and its intrinsic dimension with exact guarantees simultaneously, unifying two longstanding goals of unsupervised learning.

In this paper we demonstrate how sub-Riemannian geometry can be used for manifold learning and surface reconstruction by combining local linear approximations of a point cloud to obtain lower dimensional bundles. Local approximations obtained by local PCAs are collected into a rank k tangent subbundle on R^d, k<d, which we call a principal subbundle. This determines a sub-Riemannian metric on R^d. We show that sub-Riemannian geodesics with respect to this metric can successfully be applied to a number of important problems, such as: explicit construction of an approximating submanifold M, construction of a representation of the point-cloud in R^k, and computation of distances between observations, taking the learned geometry into account. The reconstruction is guaranteed to equal the true submanifold in the limit case where tangent spaces are estimated exactly. Via simulations, we show that the framework is robust when applied to noisy data. Furthermore, the framework generalizes to observations on an a priori known Riemannian manifold.

Sparse Autoencoders (SAEs) are widely used for mechanistic interpretability in large language models, yet their formulation assigns each latent feature a single decoder direction, implicitly assuming features to be one-dimensional. We show that this assumption mismatches with the multi-dimensional structure of model features, provably inducing feature splitting through two distinct mechanisms. Geometrically, reconstructing a feature of intrinsic dimension d_i ge 2 to error varepsilon with single-direction decoders forces a number of atoms that is exponential in d_i. From an end-to-end optimization perspective, this splitting is not merely possible but actively preferred. We prove that there exists a continuous path from the true d_i-dimensional basis to a strictly lower risk of the ell_1-regularized SAE objective, whose descent directions drive any trained dictionary into that exponential regime. A single coherent feature is therefore fragmented across many near-collinear latents, producing spurious multiplicity and obscuring the intrinsic geometry. Motivated by this, we introduce Subspace-Aware Sparse Autoencoders (SASA), which replace single-vector decoders with learned decoder subspaces, enforce block sparsity via Top-s group gating, and adapt each group's effective rank with a nuclear-norm regularizer. We then show that once the block size satisfies r ge d_i, a single group not only can represent the entire feature slice but is the global minimizer of the SASA objective. This consolidation yields a sample complexity polynomial in d_i rather than exponential -- a decisive advantage given that every training activation costs an LLM forward pass. Empirically, on GPT-2 and Mistral-7B, SASA reduces feature splitting and absorption, improves monosemanticity and interpretability, and matches or exceeds standard SAEs while training on roughly half the token budget.

Deep subspace clustering (DSC) algorithms face several challenges that hinder their widespread adoption across variois application domains. First, clustering quality is typically assessed using only the encoder's output layer, disregarding valuable information present in the intermediate layers. Second, most DSC approaches treat representation learning and subspace clustering as independent tasks, limiting their effectiveness. Third, they assume the availability of a held-out dataset for hyperparameter tuning, which is often impractical in real-world scenarios. Fourth, learning termination is commonly based on clustering error monitoring, requiring external labels. Finally, their performance often depends on post-processing techniques that rely on labeled data. To address this limitations, we introduce a novel single-view DSC approach that: (i) minimizes a layer-wise self expression loss using a joint representation matrix; (ii) optimizes a subspace-structured norm to enhance clustering quality; (iii) employs a multi-stage sequential learning framework, consisting of pre-training and fine-tuning, enabling the use of multiple regularization terms without hyperparameter tuning; (iv) incorporates a relative error-based self-stopping mechanism to terminate training without labels; and (v) retains a fixed number of leading coefficients in the learned representation matrix based on prior knowledge. We evaluate the proposed method on six datasets representing faces, digits, and objects. The results show that our method outperforms most linear SC algorithms with careffulyl tuned hyperparameters while maintaining competitive performance with the best performing linear appoaches.

This paper presents a new, provably-convergent algorithm for computing the flag-mean and flag-median of a set of points on a flag manifold under the chordal metric. The flag manifold is a mathematical space consisting of flags, which are sequences of nested subspaces of a vector space that increase in dimension. The flag manifold is a superset of a wide range of known matrix spaces, including Stiefel and Grassmanians, making it a general object that is useful in a wide variety computer vision problems. To tackle the challenge of computing first order flag statistics, we first transform the problem into one that involves auxiliary variables constrained to the Stiefel manifold. The Stiefel manifold is a space of orthogonal frames, and leveraging the numerical stability and efficiency of Stiefel-manifold optimization enables us to compute the flag-mean effectively. Through a series of experiments, we show the competence of our method in Grassmann and rotation averaging, as well as principal component analysis. We release our source code under https://github.com/nmank/FlagAveraging.

Let F be a field, and n geq r>0 be integers, with r even. Denote by A_n(F) the space of all n-by-n alternating matrices with entries in F. We consider the problem of determining the greatest possible dimension for an affine subspace of A_n(F) in which every matrix has rank equal to r (or rank at least r). Recently Rubei has solved this problem over the field of real numbers. We extend her result to all fields with large enough cardinality. Provided that n geq r+3 and |F|geq minbigl(r-1,r{2}+2bigr), we also determine the affine subspaces of rank r matrices in A_n(F) that have the greatest possible dimension, and we point to difficulties for the corresponding problem in the case nleq r+2.

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.

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

In this paper, we study the problem of principal component analysis with generative modeling assumptions, adopting a general model for the observed matrix that encompasses notable special cases, including spiked matrix recovery and phase retrieval. The key assumption is that the underlying signal lies near the range of an L-Lipschitz continuous generative model with bounded k-dimensional inputs. We propose a quadratic estimator, and show that it enjoys a statistical rate of order frac{klog L{m}}, where m is the number of samples. We also provide a near-matching algorithm-independent lower bound. Moreover, we provide a variant of the classic power method, which projects the calculated data onto the range of the generative model during each iteration. We show that under suitable conditions, this method converges exponentially fast to a point achieving the above-mentioned statistical rate. We perform experiments on various image datasets for spiked matrix and phase retrieval models, and illustrate performance gains of our method to the classic power method and the truncated power method devised for sparse principal component analysis.

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.

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.

A principal component analysis of the TCGA data for 15 cancer localizations unveils the following qualitative facts about tumors: 1) The state of a tissue in gene expression space may be described by a few variables. In particular, there is a single variable describing the progression from a normal tissue to a tumor. 2) Each cancer localization is characterized by a gene expression profile, in which genes have specific weights in the definition of the cancer state. There are no less than 2500 differentially-expressed genes, which lead to power-like tails in the expression distribution functions. 3) Tumors in different localizations share hundreds or even thousands of differentially expressed genes. There are 6 genes common to the 15 studied tumor localizations. 4) The tumor region is a kind of attractor. Tumors in advanced stages converge to this region independently of patient age or genetic variability. 5) There is a landscape of cancer in gene expression space with an approximate border separating normal tissues from tumors.

Large Language Models (LLMs) rely on safety alignment to produce socially acceptable responses. This is typically achieved through instruction tuning and reinforcement learning from human feedback. However, this alignment is known to be brittle: further fine-tuning, even on benign or lightly contaminated data, can degrade safety and reintroduce harmful behaviors. A growing body of work suggests that alignment may correspond to identifiable geometric directions in weight space, forming subspaces that could, in principle, be isolated or preserved to defend against misalignment. In this work, we conduct a comprehensive empirical study of this geometric perspective. We examine whether safety-relevant behavior is concentrated in specific subspaces, whether it can be separated from general-purpose learning, and whether harmfulness arises from distinguishable patterns in internal representations. Across both parameter and activation space, our findings are consistent: subspaces that amplify safe behaviors also amplify unsafe ones, and prompts with different safety implications activate overlapping representations. We find no evidence of a subspace that selectively governs safety. These results challenge the assumption that alignment is geometrically localized. Rather than residing in distinct directions, safety appears to emerge from entangled, high-impact components of the model's broader learning dynamics. This suggests that subspace-based defenses may face fundamental limitations and underscores the need for alternative strategies to preserve alignment under continued training. We corroborate these findings through multiple experiments on five open-source LLMs. Our code is publicly available at: https://github.com/CERT-Lab/safety-subspaces.

Deep neural networks perform remarkably well on image classification tasks but remain vulnerable to carefully crafted adversarial perturbations. This work revisits linear dimensionality reduction as a simple, data-adapted defense. We empirically compare standard Principal Component Analysis (PCA) with its sparse variant (SPCA) as front-end feature extractors for downstream classifiers, and we complement these experiments with a theoretical analysis. On the theory side, we derive exact robustness certificates for linear heads applied to SPCA features: for both ell_infty and ell_2 threat models (binary and multiclass), the certified radius grows as the dual norms of W^top u shrink, where W is the projection and u the head weights. We further show that for general (non-linear) heads, sparsity reduces operator-norm bounds through a Lipschitz composition argument, predicting lower input sensitivity. Empirically, with a small non-linear network after the projection, SPCA consistently degrades more gracefully than PCA under strong white-box and black-box attacks while maintaining competitive clean accuracy. Taken together, the theory identifies the mechanism (sparser projections reduce adversarial leverage) and the experiments verify that this benefit persists beyond the linear setting. Our code is available at https://github.com/killian31/SPCARobustness.

Vision-Language Models require efficient adaptation to continually emerging downstream tasks. While Parameter-Efficient Fine-Tuning mitigates catastrophic forgetting, assigning isolated modules per task leads to parameter explosion. Conversely, recent similarity-driven sharing mechanisms falsely equate superficial visual similarity with underlying alignment consistency. This fundamental mismatch triggers severe negative transfer between visually similar but logically distinct tasks and fails to exploit alignment reuse across visually diverse ones. We argue thatalignment sharing is fundamentally a geometric problem of overlapping optimization trajectories within shared low-rank subspaces. Grounded in this insight, we propose iGSP, a novel framework that achieves efficient adaptation via implicit gradient subspace projection. Leveraging the early convergence of MoE routers to establish the subspace basis, iGSP bifurcates the adaptation process into two phases. First, the Subspace Identification phase introduces candidate experts via basis pre-expansion, applies a novel subspace-constrained regularization to implicitly project new task gradients onto the historical subspace, and precisely prunes redundant dimensions by treating routing probabilities as gradient flow indicators, ultimately to maximize knowledge reuse. Second, the Orthogonal Subspace Fine-Tuning phase fixes this structural basis and removes the regularization to rapidly fit the task-specific residual loss. Extensive experiments on the MTIL benchmark demonstrate that iGSP achieves state-of-the-art accuracy while significantly improving training efficiency, reducing the average trainable parameters by 42.7\% compared to current SOTA methods, and decreasing the final total parameters by 86.9\% relative to counterparts. The source code is available at https://github.com/GeoX-Lab/iGSP.

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.

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.

In this paper, we study the ideas of composition and decomposition in the context of vector spaces, graphs and matroids. For vector spaces V_{AB}, treated as collection of row vectors, with specified column set Auplus B, we define V_{SP}lrarv V_{PQ}, Scap Q= emptyset, to be the collection of all vectors (f_S,f_Q) such that (f_S,f_P)in V_{SP}, (f_P,f_Q)in V_{PQ}. An analogous operation G_{SP}lrarg G_{PQ}equivd G_{PQ} can be defined in relation to graphs G_{SP}, G_{PQ}, on edge sets Suplus P, Puplus Q, respectively in terms of an overlapping subgraph G_P which gets deleted in the right side graph (see for instance the notion of k-sum oxley). For matroids we define the `linking' M_{SP}lrarm M_{PQ} equivd (M_{SP}vee M_{PQ})times (Suplus Q), denoting the contraction operation by 'times'. In each case, we examine how to minimize the size of the `overlap' set P, without affecting the right side entity. In the case of vector spaces, there is a polynomial time algorithm for achieving the minimum, which we present. Similar ideas work for graphs and for matroids under appropriate conditions. Next we consider the problem of decomposition. Here, in the case of vector spaces, the problem is to decompose V_{SQ} as V_{SP}lrarv V_{PQ}, with minimum size P. We give a polynomial time algorithm for this purpose. In the case of graphs and matroids we give a solution to this problem under certain restrictions.

Subspace clustering seeks to identify subspaces that segment a set of n data points into k (k<<n) groups, which has emerged as a powerful tool for analyzing data from various domains, especially images and videos. Recently, several studies have demonstrated the great potential of subspace clustering models for partitioning vertices in attributed graphs, referred to as SCAG. However, these works either demand significant computational overhead for constructing the nxn self-expressive matrix, or fail to incorporate graph topology and attribute data into the subspace clustering framework effectively, and thus, compromise result quality. Motivated by this, this paper presents two effective and efficient algorithms, S2CAG and M-S2CAG, for SCAG computation. Particularly, S2CAG obtains superb performance through three major contributions. First, we formulate a new objective function for SCAG with a refined representation model for vertices and two non-trivial constraints. On top of that, an efficient linear-time optimization solver is developed based on our theoretically grounded problem transformation and well-thought-out adaptive strategy. We then conduct an in-depth analysis to disclose the theoretical connection of S2CAG to conductance minimization, which further inspires the design of M-S2CAG that maximizes the modularity. Our extensive experiments, comparing S2CAG and M-S2CAG against 17 competitors over 8 benchmark datasets, exhibit that our solutions outperform all baselines in terms of clustering quality measured against the ground truth while delivering high efficiency

One technique to visualize the training of neural networks is to perform PCA on the parameters over the course of training and to project to the subspace spanned by the first few PCA components. In this paper we compare this technique to the PCA of a high dimensional random walk. We compute the eigenvalues and eigenvectors of the covariance of the trajectory and prove that in the long trajectory and high dimensional limit most of the variance is in the first few PCA components, and that the projection of the trajectory onto any subspace spanned by PCA components is a Lissajous curve. We generalize these results to a random walk with momentum and to an Ornstein-Uhlenbeck processes (i.e., a random walk in a quadratic potential) and show that in high dimensions the walk is not mean reverting, but will instead be trapped at a fixed distance from the minimum. We finally compare the distribution of PCA variances and the PCA projected training trajectories of a linear model trained on CIFAR-10 and ResNet-50-v2 trained on Imagenet and find that the distribution of PCA variances resembles a random walk with drift.

Feature-map knowledge distillation (KD) transfers internal representations well between comparably sized Vision Transformers (ViTs), but it often fails in compression. We revisit this failure and uncover a paradox. Sample-wise SVD shows that each image is highly compressible, which seems to suggest that a narrow student with a linear projector should match the teacher "in principle". However, a dataset-level view contradicts this intuition: PCA shows that the teacher is a union of low-rank subspaces with significant subspace rotation across inputs. We further introduce token-level Spectral Energy Patterns (SEP) and find an architecture-invariant encoding law: tokens spread energy broadly across channel modes even when they live in low-rank subspace, creating a bandwidth mismatch. We refer to this combined phenomenon as an encoding mismatch. We propose two minimal remedies, Lift or WideLast: (i) Lift retains a lightweight lifting projector at inference to provide wider channel, or (ii) WideLast widens only the student's last block, enabling an input-dependent expansion. On ImageNet-1K, these fixes revive feature KD for ViT compression, improving DeiT-Tiny distilled from CaiT-S24 from 74.86% to 77.53%/78.23% top-1 accuracy, and they also strengthen students trained without distillation. Our analyses clarify when and why feature-map KD fails and then how to fix it. Code and raw data are provided in https://github.com/thy960112/From-Per-Image-Low-Rank-to-Encoding-Mismatch.

Recent unsupervised domain adaptation methods have utilized vicinal space between the source and target domains. However, the equilibrium collapse of labels, a problem where the source labels are dominant over the target labels in the predictions of vicinal instances, has never been addressed. In this paper, we propose an instance-wise minimax strategy that minimizes the entropy of high uncertainty instances in the vicinal space to tackle the stated problem. We divide the vicinal space into two subspaces through the solution of the minimax problem: contrastive space and consensus space. In the contrastive space, inter-domain discrepancy is mitigated by constraining instances to have contrastive views and labels, and the consensus space reduces the confusion between intra-domain categories. The effectiveness of our method is demonstrated on public benchmarks, including Office-31, Office-Home, and VisDA-C, achieving state-of-the-art performances. We further show that our method outperforms the current state-of-the-art methods on PACS, which indicates that our instance-wise approach works well for multi-source domain adaptation as well. Code is available at https://github.com/NaJaeMin92/CoVi.

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.

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.

Large language models (LLMs) handle many tasks with one set of parameters, but under KV-cached inference it is unclear what task-general structure, if any, is used at decode time rather than during prefill. We propose DecodeShare, a protocol that identifies a low-dimensional subspace consistently shared across tasks in decode-time hidden states, and then tests its causal role by removing that subspace only during decoding. In our experiments, disturbing the discovered shared subspace degrades decision performance far more than disturbing either a prefill-derived or random subspace under the same intervention budget. We further show this decode-shared subspace has practical consequences for activation steering: common steering directions can overlap the task-general decode channel. Projecting out this shared subspace directly separates the functional roles of the two components, while evaluating steering vectors at decode-time yields more reliable signal for downstream deployment than prefill-based proxies. Despite its compactness, the shared subspace can serve as a high-leverage causal channel at decode time. Code is available at: https://github.com/Zishan-Shao/decodeshare.git.

Krylov subspace, which is generated by multiplying a given vector by the matrix of a linear transformation and its successive powers, has been extensively studied in classical optimization literature to design algorithms that converge quickly for large linear inverse problems. For example, the conjugate gradient method (CG), one of the most popular Krylov subspace methods, is based on the idea of minimizing the residual error in the Krylov subspace. However, with the recent advancement of high-performance diffusion solvers for inverse problems, it is not clear how classical wisdom can be synergistically combined with modern diffusion models. In this study, we propose a novel and efficient diffusion sampling strategy that synergistically combines the diffusion sampling and Krylov subspace methods. Specifically, we prove that if the tangent space at a denoised sample by Tweedie's formula forms a Krylov subspace, then the CG initialized with the denoised data ensures the data consistency update to remain in the tangent space. This negates the need to compute the manifold-constrained gradient (MCG), leading to a more efficient diffusion sampling method. Our method is applicable regardless of the parametrization and setting (i.e., VE, VP). Notably, we achieve state-of-the-art reconstruction quality on challenging real-world medical inverse imaging problems, including multi-coil MRI reconstruction and 3D CT reconstruction. Moreover, our proposed method achieves more than 80 times faster inference time than the previous state-of-the-art method. Code is available at https://github.com/HJ-harry/DDS

Modern transformers pair impressive capabilities with substantial memory and compute demands. Low-rank weight factorization reduces both while keeping the matrices dense, and thus efficient on standard hardware. Existing methods, however, choose the subspace to remove from each weight matrix with local closed-form criteria: activation energy, layer-wise reconstruction error, or a quadratic approximation of the loss. These criteria ignore how errors propagate through the network, so at high compression the errors compound with depth and performance collapses. We introduce Learnable Subspace Projections (LSP), which instead learns the subspaces to discard end-to-end. Each linear layer, or tied group of layers that read the same activations, is assigned an orthogonal projector. All projectors are optimized jointly against a global objective--the KL divergence to the dense model's output distribution or the model's original training loss--while the pretrained weights remain frozen. Projectors are initialized from a whitened SVD truncation, and ranks are allocated by the output KL each projector induces per parameter saved. After training, the projectors merge into standard low-rank factors, with each tied group sharing one factor. In attention, this also lets the model cache one narrow latent in place of full keys and values. Across LLMs (OPT-125M/1.3B, Qwen3-4B, Llama-2-7B) and ViT-B/16, LSP outperforms baselines, and its advantage widens as compression increases. At -70% compression, LSP brings Llama-2-7B to 10.9 WikiText-2 perplexity and 42.2% mean zero-shot accuracy, versus 13.3 and 36.0% for the strongest baseline. The factorized model decodes up to 1.6x faster than the dense model at small batch sizes, and aching the shared latent shrinks the combined memory of weights and KV cache by 13.5x at a 128k-token context, versus at most 6.5x for untied baseline factorizations.

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.

The recent introduction of synthetic correlated diffusion (CDI^s) imaging has demonstrated significant potential in the realm of clinical decision support for prostate cancer (PCa). CDI^s is a new form of magnetic resonance imaging (MRI) designed to characterize tissue characteristics through the joint correlation of diffusion signal attenuation across different Brownian motion sensitivities. Despite the performance improvement, the CDI^s data for PCa has not been previously made publicly available. In our commitment to advance research efforts for PCa, we introduce Cancer-Net PCa-Data, an open-source benchmark dataset of volumetric CDI^s imaging data of PCa patients. Cancer-Net PCa-Data consists of CDI^s volumetric images from a patient cohort of 200 patient cases, along with full annotations (gland masks, tumor masks, and PCa diagnosis for each tumor). We also analyze the demographic and label region diversity of Cancer-Net PCa-Data for potential biases. Cancer-Net PCa-Data is the first-ever public dataset of CDI^s imaging data for PCa, and is a part of the global open-source initiative dedicated to advancement in machine learning and imaging research to aid clinicians in the global fight against cancer.

Quantum computing is powerful because unitary operators describing the time-evolution of a quantum system have exponential size in terms of the number of qubits present in the system. We develop a new "Singular value transformation" algorithm capable of harnessing this exponential advantage, that can apply polynomial transformations to the singular values of a block of a unitary, generalizing the optimal Hamiltonian simulation results of Low and Chuang. The proposed quantum circuits have a very simple structure, often give rise to optimal algorithms and have appealing constant factors, while usually only use a constant number of ancilla qubits. We show that singular value transformation leads to novel algorithms. We give an efficient solution to a certain "non-commutative" measurement problem and propose a new method for singular value estimation. We also show how to exponentially improve the complexity of implementing fractional queries to unitaries with a gapped spectrum. Finally, as a quantum machine learning application we show how to efficiently implement principal component regression. "Singular value transformation" is conceptually simple and efficient, and leads to a unified framework of quantum algorithms incorporating a variety of quantum speed-ups. We illustrate this by showing how it generalizes a number of prominent quantum algorithms, including: optimal Hamiltonian simulation, implementing the Moore-Penrose pseudoinverse with exponential precision, fixed-point amplitude amplification, robust oblivious amplitude amplification, fast QMA amplification, fast quantum OR lemma, certain quantum walk results and several quantum machine learning algorithms. In order to exploit the strengths of the presented method it is useful to know its limitations too, therefore we also prove a lower bound on the efficiency of singular value transformation, which often gives optimal bounds.

Robust principal component analysis (RPCA) decomposes an observation matrix into low-rank background and sparse object components. This capability has enabled its application in tasks ranging from image restoration to segmentation. However, traditional RPCA models suffer from computational burdens caused by matrix operations, reliance on finely tuned hyperparameters, and rigid priors that limit adaptability in dynamic scenarios. To solve these limitations, we propose RPCANet++, a sparse object segmentation framework that fuses the interpretability of RPCA with efficient deep architectures. Our approach unfolds a relaxed RPCA model into a structured network comprising a Background Approximation Module (BAM), an Object Extraction Module (OEM), and an Image Restoration Module (IRM). To mitigate inter-stage transmission loss in the BAM, we introduce a Memory-Augmented Module (MAM) to enhance background feature preservation, while a Deep Contrast Prior Module (DCPM) leverages saliency cues to expedite object extraction. Extensive experiments on diverse datasets demonstrate that RPCANet++ achieves state-of-the-art performance under various imaging scenarios. We further improve interpretability via visual and numerical low-rankness and sparsity measurements. By combining the theoretical strengths of RPCA with the efficiency of deep networks, our approach sets a new baseline for reliable and interpretable sparse object segmentation. Codes are available at our Project Webpage https://fengyiwu98.github.io/rpcanetx.

The singular value decomposition (SVD) is a powerful tool in modern numerical linear algebra, which underpins computational methods such as principal component analysis (PCA), low-rank approximations, and randomized algorithms. Many practical scenarios require solving numerous small SVD problems, a regime generally referred to as "batch SVD". Existing programming models can handle this efficiently on parallel CPU architectures, but high-performance solutions for GPUs remain immature. A GPU-oriented batch SVD solver is introduced. This solver exploits the one-sided Jacobi algorithm to exploit fine-grained parallelism, and a number of algorithmic and design optimizations achieve unmatched performance. Starting from a baseline solver, a sequence of optimizations is applied to obtain incremental performance gains. Numerical experiments show that the new solver is robust across problems with different numerical properties, matrix shapes, and arithmetic precisions. Performance benchmarks on both NVIDIA and AMD systems show significant performance speedups over vendor solutions as well as existing open-source solvers.

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.

We present an algorithm for decomposing low rank tensors of any symmetry type, from fully asymmetric to fully symmetric. It generalizes the recent subspace power method from symmetric tensors to all tensors. The algorithm transforms an input tensor into a tensor with orthonormal slices. We show that for tensors with orthonormal slices and low rank, the summands of their decomposition are in one-to-one correspondence with the partially symmetric singular vector tuples (pSVTs) with singular value one. We use this to show correctness of the algorithm. We introduce a shifted power method for computing pSVTs and establish its global convergence. Numerical experiments demonstrate that our decomposition algorithm achieves higher accuracy and faster runtime than existing methods.

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.

Dimensionality reduction is a fundamental task that aims to simplify complex data by reducing its feature dimensionality while preserving essential patterns, with core applications in data analysis and visualisation. To preserve the underlying data structure, multi-dimensional scaling (MDS) methods focus on preserving pairwise dissimilarities, such as distances. They optimise the embedding to have pairwise distances as close as possible to the data dissimilarities. However, the current standard is limited to embedding data in Riemannian manifolds. Motivated by the lack of asymmetry in the Riemannian metric of the embedding space, this paper extends the MDS problem to a natural asymmetric generalisation of Riemannian manifolds called Finsler manifolds. Inspired by Euclidean space, we define a canonical Finsler space for embedding asymmetric data. Due to its simplicity with respect to geodesics, data representation in this space is both intuitive and simple to analyse. We demonstrate that our generalisation benefits from the same theoretical convergence guarantees. We reveal the effectiveness of our Finsler embedding across various types of non-symmetric data, highlighting its value in applications such as data visualisation, dimensionality reduction, directed graph embedding, and link prediction.

Low-Rank Adaptation (LoRA) merging can efficiently combine diverse generative capabilities from multiple trained LoRAs for a diffusion model. However, existing LoRA merging techniques often suffer from severe parameter interference, causing destructive collisions in the shared parameter space. To address this, we propose Subspace Signal Routing (SSR), which resolves interference by routing internal signals instead of performing parameter-space merge. Specifically, SSR first constructs a unified subspace by concatenating candidate LoRAs along the rank dimension. Next, SSR employs an inverse correlation matrix to decorrelate mixed signals within this space. Finally, a directional guide matrix steers these purified signals into their respective task-specific subspaces. We provide a rigorous theoretical analysis proving that SSR aligns with the Ordinary Least Squares (OLS) solution, thereby ensuring mathematical optimality. We utilize the additivity of sufficient statistics to design a streaming algorithm. This enables on-the-fly updates that significantly reduce memory overhead and computation time. Extensive experiments validate that SSR significantly outperforms state-of-the-art methods while maintaining comparable efficiency. Code is available at https://github.com/nagara214/SSR-Merge.

This paper tackles the data-driven approximation of unknown dynamical systems using Koopman-operator methods. Given a dictionary of functions, these methods approximate the projection of the action of the operator on the finite-dimensional subspace spanned by the dictionary. We propose the Tunable Symmetric Subspace Decomposition algorithm to refine the dictionary, balancing its expressiveness and accuracy. Expressiveness corresponds to the ability of the dictionary to describe the evolution of as many observables as possible and accuracy corresponds to the ability to correctly predict their evolution. Based on the observation that Koopman-invariant subspaces give rise to exact predictions, we reason that prediction accuracy is a function of the degree of invariance of the subspace generated by the dictionary and provide a data-driven measure to measure invariance proximity. The proposed algorithm iteratively prunes the initial functional space to identify a refined dictionary of functions that satisfies the desired level of accuracy while retaining as much of the original expressiveness as possible. We provide a full characterization of the algorithm properties and show that it generalizes both Extended Dynamic Mode Decomposition and Symmetric Subspace Decomposition. Simulations on planar systems show the effectiveness of the proposed methods in producing Koopman approximations of tunable accuracy that capture relevant information about the dynamical system.

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.

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.

Mixture of Experts (MoE) models scale capacity but often suffer from representation collapse and gradient instability. We propose Dynamic Subspace Composition (DSC), a framework that approximates context-dependent weights via a state-dependent, sparse expansion of a shared basis bank. Formally, DSC models the weight update as a residual trajectory within a Star- Shaped Domain, employing a Magnitude-Gated Simplex Interpolation to ensure continuity at the identity. Unlike standard Mixture-of-LoRAs, which incurs O(M rd) parameter complexity by retrieving independent rank-r matrices, DSC constructs a compositional rank-K approximation from decoupled unit-norm basis vectors. This reduces parameter complexity to O(M d) and memory traffic to O(Kd), while Frame-Theoretic regularization and spectral constraints provide rigorous worst-case bounds on the dynamic update. The code is available at https://github. com/VladimerKhasia/DSC

Retrieval-Augmented Generation (RAG) has emerged as a powerful paradigm for grounding large language models in external knowledge sources, improving the precision of agents responses. However, high-dimensional language model embeddings, often in the range of hundreds to thousands of dimensions, can present scalability challenges in terms of storage and latency, especially when processing massive financial text corpora. This paper investigates the use of Principal Component Analysis (PCA) to reduce embedding dimensionality, thereby mitigating computational bottlenecks without incurring large accuracy losses. We experiment with a real-world dataset and compare different similarity and distance metrics under both full-dimensional and PCA-compressed embeddings. Our results show that reducing vectors from 3,072 to 110 dimensions provides a sizeable (up to 60times) speedup in retrieval operations and a sim 28.6times reduction in index size, with only moderate declines in correlation metrics relative to human-annotated similarity scores. These findings demonstrate that PCA-based compression offers a viable balance between retrieval fidelity and resource efficiency, essential for real-time systems such as Zanista AI's Newswitch platform. Ultimately, our study underscores the practicality of leveraging classical dimensionality reduction techniques to scale RAG architectures for knowledge-intensive applications in finance and trading, where speed, memory efficiency, and accuracy must jointly be optimized.

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 study principal-agent problems in which a principal commits to an outcome-dependent payment scheme -- called contract -- in order to induce an agent to take a costly, unobservable action leading to favorable outcomes. We consider a generalization of the classical (single-round) version of the problem in which the principal interacts with the agent by committing to contracts over multiple rounds. The principal has no information about the agent, and they have to learn an optimal contract by only observing the outcome realized at each round. We focus on settings in which the size of the agent's action space is small. We design an algorithm that learns an approximately-optimal contract with high probability in a number of rounds polynomial in the size of the outcome space, when the number of actions is constant. Our algorithm solves an open problem by Zhu et al.[2022]. Moreover, it can also be employed to provide a mathcal{O}(T^{4/5}) regret bound in the related online learning setting in which the principal aims at maximizing their cumulative utility, thus considerably improving previously-known regret bounds.

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.

We analyze the stability of a class of eigensolvers that target interior eigenvalues with rational filters. We show that subspace iteration with a rational filter is robust even when an eigenvalue is near a filter's pole. These dangerous eigenvalues contribute to large round-off errors in the first iteration, but are self-correcting in later iterations. For matrices with orthogonal eigenvectors (e.g., real-symmetric or complex Hermitian), two iterations is enough to reduce round-off errors to the order of the unit-round off. In contrast, Krylov methods accelerated by rational filters with fixed poles typically fail to converge to unit round-off accuracy when an eigenvalue is close to a pole. In the context of Arnoldi with shift-and-invert enhancement, we demonstrate a simple restart strategy that recovers full precision in the target eigenpairs.

Current unlearning and safety training methods consistently fail to remove dangerous knowledge from language models. We identify the root cause - unlearning targets representations which are too general - and develop a highly selective technique that unlearns robustly while preserving general performance. Our method performs PCA on activations and module-output gradients to identify subspaces containing common representations, then collapses these subspaces before computing unlearning updates, a technique we term Collapse of Irrelevant Representations (CIR). This avoids unlearning general knowledge and targets only representations specific to the facts being unlearned. When unlearning bio- and cyber-hazardous facts from Llama-3.1-8B, we achieve over 30x greater reduction in post-attack accuracy than the best baseline (Circuit Breakers), while disrupting general performance 30x less, and using less than 3 GPU-seconds per fact. Thus, by disentangling harmful and benign capabilities at the level of representations, CIR enables robust and non-disruptive unlearning.

This paper introduces Least Volume-a simple yet effective regularization inspired by geometric intuition-that can reduce the necessary number of latent dimensions needed by an autoencoder without requiring any prior knowledge of the intrinsic dimensionality of the dataset. We show that the Lipschitz continuity of the decoder is the key to making it work, provide a proof that PCA is just a linear special case of it, and reveal that it has a similar PCA-like importance ordering effect when applied to nonlinear models. We demonstrate the intuition behind the regularization on some pedagogical toy problems, and its effectiveness on several benchmark problems, including MNIST, CIFAR-10 and CelebA.

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.

Spectral positional encodings (PEs) for directed graphs face two obstacles: magnetic Laplacians require an O(n^3) Hermitian eigendecomposition per potential, and their complex eigenvectors are defined only up to unitary gauge, which prior work handles with basis-invariant architectures. We propose learnable spectral PEs of the form h_θ(A_q),R, where A_q is a normalized magnetic operator, h_θ a learnable scalar spectral response, and R a block of random probes. Because the PE is a matrix function of the operator, it is gauge-invariant by construction. We compute it in a Hermitian block Krylov subspace from sparse matrix--vector products only, prove that k = O(log(1/varepsilon)) block steps suffice uniformly over heat--resolvent response families, and give a covering-number argument for why low-dimensional structured families generalize where free per-eigenvalue weights overfit. On a directed SBM whose symmetrization is uninformative by construction, direction-blind PEs stay at chance while magnetic Krylov PEs converge to the exact-eigendecomposition oracle as the depth grows. The same probes yield gauge-invariant pairwise features with 1/s Monte-Carlo error, and the undirected q{=}0 case improves heterophilous benchmarks over no-PE and polynomial baselines.

Model merging enables the combination of multiple specialized expert models into a single model capable of performing multiple tasks. However, the benefits of merging an increasing amount of specialized experts generally lead to diminishing returns and reduced overall performance gains. In this work, we empirically and theoretically analyze this limitation, proving that for Task Arithmetic-based methods, as more experts are merged, the common information dominates the task-specific information, leading to inevitable rank collapse. To mitigate this issue, we introduce Subspace Boosting, which operates on the singular value decomposed task vector space and maintains task vector ranks. Subspace Boosting raises merging efficacy for up to 20 experts by large margins of more than 10% when evaluated on both vision and language benchmarks. Moreover, we propose employing Higher-Order Generalized Singular Value Decomposition to quantify task similarity, offering a new interpretable perspective on model merging. Code and models are available at https://github.com/ronskoro/Subspace-Boosting.

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.

Sparse autoencoders (SAEs) have emerged as a promising method for interpreting neural network representations by decomposing activations into sparse combinations of dictionary atoms. However, SAEs assume that features combine additively through linear reconstruction, an assumption that cannot capture compositional structure: linear models cannot distinguish whether "Starbucks" arises from the composition of "star" and "coffee" features or merely their co-occurrence. This forces SAEs to allocate monolithic features for compound concepts rather than decomposing them into interpretable constituents. We introduce PolySAE, which extends the SAE decoder with higher-order terms to model feature interactions while preserving the linear encoder essential for interpretability. Through low-rank tensor factorization on a shared projection subspace, PolySAE captures pairwise and triple feature interactions with small parameter overhead (3% on GPT2). Across four language models and three SAE variants, PolySAE achieves an average improvement of approximately 8% in probing F1 while maintaining comparable reconstruction error, and produces 2-10times larger Wasserstein distances between class-conditional feature distributions. Critically, learned interaction weights exhibit negligible correlation with co-occurrence frequency (r = 0.06 vs. r = 0.82 for SAE feature covariance), suggesting that polynomial terms capture compositional structure, such as morphological binding and phrasal composition, largely independent of surface statistics.

We propose a projection pursuit (PP) algorithm based on Gaussian mixture models (GMMs). The negentropy obtained from a multivariate density estimated by GMMs is adopted as the PP index to be maximised. For a fixed dimension of the projection subspace, the GMM-based density estimation is projected onto that subspace, where an approximation of the negentropy for Gaussian mixtures is computed. Then, Genetic Algorithms (GAs) are used to find the optimal, orthogonal projection basis by maximising the former approximation. We show that this semi-parametric approach to PP is flexible and allows highly informative structures to be detected, by projecting multivariate datasets onto a subspace, where the data can be feasibly visualised. The performance of the proposed approach is shown on both artificial and real datasets.

Differential geometric approaches to the analysis and processing of data in the form of symmetric positive definite (SPD) matrices have had notable successful applications to numerous fields including computer vision, medical imaging, and machine learning. The dominant geometric paradigm for such applications has consisted of a few Riemannian geometries associated with spectral computations that are costly at high scale and in high dimensions. We present a route to a scalable geometric framework for the analysis and processing of SPD-valued data based on the efficient computation of extreme generalized eigenvalues through the Hilbert and Thompson geometries of the semidefinite cone. We explore a particular geodesic space structure based on Thompson geometry in detail and establish several properties associated with this structure. Furthermore, we define a novel iterative mean of SPD matrices based on this geometry and prove its existence and uniqueness for a given finite collection of points. Finally, we state and prove a number of desirable properties that are satisfied by this mean.

High-dimensional vector spaces, particularly embedding spaces with dense semantic structure, are often interpreted primarily leveraging solely geometric relationships. In this work, we show that they can also be viewed as spectral energy networks induced by the topology of their underlying feature-space manifold with relevant improvements for downstream tasks. Building on this perspective, we introduce Graph Wiring, a general framework for exploiting feature-space spectral structure, together with Spectral Indexing, its task-specific instantiation for vector search. By coupling geometric similarity with spectral information, the proposed method improves Head-Tail coherence and semantic alignment relative to purely geometric retrieval methods. It further supports adaptive search behavior through tau-modulation, providing the flexibility increasingly required by modern Retrieval-Augmented Generation (RAG) pipelines. We present the complete algorithmic pipeline, establish its theoretical foundation through epiplexity, and evaluate the approach across benchmark and industrial settings using the open-source arrowspace library.

Sparse Autoencoders (SAEs) can efficiently identify candidate monosemantic features from pretrained neural networks for galaxy morphology. We demonstrate this on Euclid Q1 images using both supervised (Zoobot) and new self-supervised (MAE) models. Our publicly released MAE achieves superhuman image reconstruction performance. While a Principal Component Analysis (PCA) on the supervised model primarily identifies features already aligned with the Galaxy Zoo decision tree, SAEs can identify interpretable features outside of this framework. SAE features also show stronger alignment than PCA with Galaxy Zoo labels. Although challenges in interpretability remain, SAEs provide a powerful engine for discovering astrophysical phenomena beyond the confines of human-defined classification.

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.

In this reproduction study, we revisit recent claims that self-attention implements kernel principal component analysis (KPCA) (Teo et al., 2024), positing that (i) value vectors V capture the eigenvectors of the Gram matrix of the keys, and (ii) that self-attention projects queries onto the principal component axes of the key matrix K in a feature space. Our analysis reveals three critical inconsistencies: (1) No alignment exists between learned self-attention value vectors and what is proposed in the KPCA perspective, with average similarity metrics (optimal cosine similarity leq 0.32, linear CKA (Centered Kernel Alignment) leq 0.11, kernel CKA leq 0.32) indicating negligible correspondence; (2) Reported decreases in reconstruction loss J_proj, arguably justifying the claim that the self-attention minimizes the projection error of KPCA, are misinterpreted, as the quantities involved differ by orders of magnitude (sim!10^3); (3) Gram matrix eigenvalue statistics, introduced to justify that V captures the eigenvector of the gram matrix, are irreproducible without undocumented implementation-specific adjustments. Across 10 transformer architectures, we conclude that the KPCA interpretation of self-attention lacks empirical support.

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.

KV cache has become a de facto technique for the inference of large language models (LLMs), where tensors of shape (layer number, head number, sequence length, feature dimension) are introduced to cache historical information for self-attention. As the size of the model and data grows, the KV cache can quickly become a bottleneck within the system in both storage and memory transfer. To address this, prior studies usually focus on the first three axes of the cache tensors for compression. This paper supplements them, focusing on the feature dimension axis, by utilizing low-rank projection matrices to transform the cache features into spaces with reduced dimensions. We begin by investigating the canonical orthogonal projection method for data compression through principal component analysis (PCA). We observe the issue with PCA projection where significant performance degradation is observed at low compression rates. To bridge the gap, we propose to directly tune the orthogonal projection matrices with a distillation objective using an elaborate Matryoshka training strategy. After training, we adaptively search for the optimal compression rates for various layers and heads given varying compression budgets. Compared to previous works, our method can easily embrace pre-trained LLMs and hold a smooth tradeoff between performance and compression rate. We empirically witness the high data efficiency of our training procedure and find that our method can sustain over 90% performance with an average KV cache compression rate of 60% (and up to 75% in certain extreme scenarios) for popular LLMs like LLaMA2-7B-base and Mistral-7B-v0.3-base.

Providing a model that achieves a strong predictive performance and is simultaneously interpretable by humans is one of the most difficult challenges in machine learning research due to the conflicting nature of these two objectives. To address this challenge, we propose a modification of the radial basis function neural network model by equipping its Gaussian kernel with a learnable precision matrix. We show that precious information is contained in the spectrum of the precision matrix that can be extracted once the training of the model is completed. In particular, the eigenvectors explain the directions of maximum sensitivity of the model revealing the active subspace and suggesting potential applications for supervised dimensionality reduction. At the same time, the eigenvectors highlight the relationship in terms of absolute variation between the input and the latent variables, thereby allowing us to extract a ranking of the input variables based on their importance to the prediction task enhancing the model interpretability. We conducted numerical experiments for regression, classification, and feature selection tasks, comparing our model against popular machine learning models, the state-of-the-art deep learning-based embedding feature selection techniques, and a transformer model for tabular data. Our results demonstrate that the proposed model does not only yield an attractive prediction performance compared to the competitors but also provides meaningful and interpretable results that potentially could assist the decision-making process in real-world applications. A PyTorch implementation of the model is available on GitHub at the following link. https://github.com/dannyzx/Gaussian-RBFNN

Targeted data selection has emerged as a crucial paradigm for efficient instruction tuning, aiming to identify a small yet influential subset of training examples for a specific target task. In practice, influence is often measured through the effect of an example on parameter updates. To make selection scalable, many approaches leverage optimizer statistics (e.g., Adam states) as an axis-aligned surrogate for update geometry (i.e., diagonal precondition), implicitly treating parameters as coordinate-wise independent. We show that this assumption breaks down in parameter-efficient fine-tuning (PEFT) methods such as LoRA. In this setting, the induced optimization geometry exhibits strong cross-parameter coupling with non-trivial off-diagonal interactions, while the task-relevant update directions are confined to a low-dimensional subspace. Motivated by this mismatch, we propose GIST (Gradient Isometric Subspace Transformation), a simple yet principled alternative that replaces axis-aligned scaling with robust subspace alignment. GIST recovers a task-specific subspace from validation gradients via spectral filtering (SVD), projects training gradients into this coupled subspace, and scores examples by their alignment with target directions.Extensive experiments have demonstrated that GIST matches or outperforms the state-of-the-art baseline with only 0.29% of the storage and 25% of the computational time under the same selection budget.

Domain Generalization (DG) aims to learn representations robust to distribution shifts. Recent geometric alignment methods, such as CPCANet, extract domain-invariant structures through batch-wise Common Principal Component Analysis (CPCA). However, CPCANet suffers from rank-deficient covariance estimation due to the small-sample-size issue in mini-batch training. To address this limitation, we propose Projection Pursuit CPCANet (PP-CPCANet), a covariance-free framework that learns a global orthogonal basis on the Stiefel manifold and jointly optimizes it with network parameters via the Cayley transform. We further introduce a symmetry-breaking detached-median PP dispersion objective to extract common principal components (CPCs) with dense and robust optimization signals. Experiments on four DG benchmarks show that PP-CPCANet achieves SOTA performance while maintaining stable training.

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.

Denoising Diffusion Models (DDMs) have emerged as a strong competitor to Generative Adversarial Networks (GANs). However, despite their widespread use in image synthesis and editing applications, their latent space is still not as well understood. Recently, a semantic latent space for DDMs, coined `h-space', was shown to facilitate semantic image editing in a way reminiscent of GANs. The h-space is comprised of the bottleneck activations in the DDM's denoiser across all timesteps of the diffusion process. In this paper, we explore the properties of h-space and propose several novel methods for finding meaningful semantic directions within it. We start by studying unsupervised methods for revealing interpretable semantic directions in pretrained DDMs. Specifically, we show that global latent directions emerge as the principal components in the latent space. Additionally, we provide a novel method for discovering image-specific semantic directions by spectral analysis of the Jacobian of the denoiser w.r.t. the latent code. Next, we extend the analysis by finding directions in a supervised fashion in unconditional DDMs. We demonstrate how such directions can be found by relying on either a labeled data set of real images or by annotating generated samples with a domain-specific attribute classifier. We further show how to semantically disentangle the found direction by simple linear projection. Our approaches are applicable without requiring any architectural modifications, text-based guidance, CLIP-based optimization, or model fine-tuning.

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.