Daily Papers

As generated images become increasingly realistic, reliable forgery detection is essential for maintaining trust in visual information. However, existing methods primarily rely on task-specific supervision to adapt vision foundation model representations, without fully exploiting internal forensic knowledge to guide detection. To address this limitation, we propose Reserve-Guided Elicitation (RGE), a framework that treats sparse, origin-sensitive internal components in pretrained models as a forensic reserve and translates their localization into structural constraints for lightweight adaptation. Specifically, we first use the Forensic Lens (F-lens) to decompose activations across layers and token groups into independent components and globally screen them by their response differences between real and generated images, identifying reserve sites and directions. Next, we map the selected directions back to hidden-state space to construct fixed reserve subspaces and insert Forensic Reserve Adapters (FRA) only at the identified sites. Finally, with the backbone parameters, previously fitted reference classifier, and subspace bases fixed, we train only the FRA coefficient maps to generate input-dependent residual updates constrained to the corresponding subspaces, strengthening existing forensic responses. Using only 500 labeled training images and a trainable parameter budget below 0.2% of the backbone, RGE achieves competitive performance across three detection benchmarks without target-benchmark adaptation. Furthermore, RGE consistently improves over the corresponding frozen detectors across eight encoders spanning self-supervised and vision-language pretraining, eliciting a latent forensic capacity broadly shared across pretrained vision models.

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.

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.

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).

We prove that: I. If L is a T_1 space, |L|>1 and d(L) leq kappa geq omega, then there is a submaximal dense subspace X of L^{2^kappa} such that |X|=Delta(X)=kappa; II. If cleqkappa=kappa^omega<lambda and 2^kappa=2^lambda, then there is a Tychonoff pseudocompact globally and locally connected space X such that |X|=Delta(X)=lambda and X is not kappa^+-resolvable; III. If omega_1leqkappa<lambda and 2^kappa=2^lambda, then there is a regular space X such that |X|=Delta(X)=lambda, all continuous real-valued functions on X are constant (so X is pseudocompact and connected) and X is not kappa^+-resolvable.

Quantum algorithms for probing ground-state properties of quantum systems require good initial states. Projection-based methods such as eigenvalue filtering rely on inputs that have a significant overlap with the low-energy subspace, which can be challenging for large, strongly-correlated systems. This issue has motivated the study of physically-inspired dynamical approaches such as thermodynamic cooling. In this work, we introduce a ground-state preparation algorithm based on the simulation of quantum dynamics. Our main insight is to transform the Hamiltonian by a shifted sign function via quantum signal processing, effectively mapping eigenvalues into positive and negative subspaces separated by a large gap. This automatically ensures that all states within each subspace conserve energy with respect to the transformed Hamiltonian. Subsequent time-evolution with a perturbed Hamiltonian induces transitions to lower-energy states while preventing unwanted jumps to higher energy states. The approach does not rely on a priori knowledge of energy gaps and requires no additional qubits to model a bath. Furthermore, it makes mathcal{O}(d^{,3/2}/epsilon) queries to the time-evolution operator of the system and mathcal{O}(d^{,3/2}) queries to a block-encoding of the perturbation, for d cooling steps and an epsilon-accurate energy resolution. Our results provide a framework for combining quantum signal processing and Hamiltonian simulation to design heuristic quantum algorithms for ground-state preparation.

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.

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.

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

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.

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

The space of unitary C_{0}-semigroups on separable infinite dimensional Hilbert space, when viewed under the topology of uniform weak convergence on compact subsets of R_{+}, is known to admit various interesting residual subspaces. Before treating the contractive case, the problem of the complete metrisability of this space was raised in [Eisner, 2010]. Utilising Borel complexity computations and automatic continuity results for semigroups, we obtain a general result, which in particular implies that the one-/multiparameter contractive C_{0}-semigroups constitute Polish spaces and thus positively addresses the open problem.

Working over infinite dimensional separable Hilbert spaces, residual results have been achieved for the space of contractive C_{0}-semigroups under the topology of uniform weak operator convergence on compact subsets of R_{+}. Eisner and Serény raised in 2009 the open problem: Does this space constitute a Baire space? Observing that the subspace of unitary semigroups is completely metrisable and appealing to known density results, we solve this problem positively by showing that certain topological properties can in general be transferred from dense subspaces to larger spaces. The transfer result in turn relies upon classification of topological properties via infinite games. Our approach is sufficiently general and can be applied to other contexts, e.g. the space of contractions under the pw-topology.

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.

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

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

We 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.

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.

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.

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.

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.