In this paper, we focus on learning product graphs from multi-domain data. We assume that the product graph is formed by the Cartesian product of two smaller graphs, which we refer to as graph factors. We pose the product graph learning problem as the problem of estimating the graph factor Laplacian matrices. To capture local interactions in data, we seek sparse graph factors and assume a smoothness model for data. We propose an efficient iterative solver for learning sparse product graphs from data. We then extend this solver to infer multi-component graph factors with applications to product graph clustering by imposing rank constraints on the graph Laplacian matrices. Although working with smaller graph factors is computationally more attractive, not all graphs may readily admit an exact Cartesian product factorization. To this end, we propose efficient algorithms to approximate a graph by a nearest Cartesian product of two smaller graphs. The efficacy of the developed framework is demonstrated using several numerical experiments on synthetic data and real data.
Learning-based low rank approximation algorithms can significantly improve the performance of randomized low rank approximation with sketch matrix. With the learned value and fixed non-zero positions for sketch matrices from learning-based algorithms, these matrices can reduce the test error of low rank approximation significantly. However, there is still no good method to learn non-zero positions as well as overcome the out-of-distribution performance loss. In this work, we introduce two new methods Learning Sparsity and Learning Randomness which try to learn a better sparsity patterns and ad…
Network-oriented research has been increasingly popular in many scientific areas. In neuroscience research, imaging-based network connectivity measures have become the key for understanding brain organizations, potentially serving as individual neural fingerprints. There are major challenges in analyzing connectivity matrices including the high dimensionality of brain networks, unknown latent sources underlying the observed connectivity, and the large number of brain connections leading to spurious findings. In this paper, we propose a novel blind source separation method with low-rank structu…
Learning sketching matrices for fast and accurate low-rank approximation (LRA) has gained increasing attention. Recently, Bartlett, Indyk, and Wagner (COLT 2022) presented a generalization bound for the learning-based LRA. Specifically, for rank-$k$ approximation using an $m \times n$ learned sketching matrix with $s$ non-zeros in each column, they proved an $\tilde{\mathrm{O}}(nsm)$ bound on the \emph{fat shattering dimension} ($\tilde{\mathrm{O}}$ hides logarithmic factors). We build on their work and make two contributions. 1. We present a better $\tilde{\mathrm{O}}(nsk)$ bound ($k \le m$).…
Zhejiang University（浙江大学）；The University of Hong Kong（香港大学）；Tongji University（同济大学）；Shenzhen Research Institute of Big Data（深圳大数据研究院）；National and Kapodistrian University of Athens（雅典国立卡波迪斯特里安大学）；Aalborg University（奥尔堡大学）
Tensor rank learning for canonical polyadic decomposition (CPD) has long been deemed as an essential yet challenging problem. In particular, since the tensor rank controls the complexity of the CPD model, its inaccurate learning would cause overfitting to noise or underfitting to the signal sources, and even destroy the interpretability of model parameters. However, the optimal determination of a tensor rank is known to be a non-deterministic polynomial-time hard (NP-hard) task. Rather than exhaustively searching for the best tensor rank via trial-and-error experiments, Bayesian inference unde…
University of Luxembourg（卢森堡大学）；Technische Universität Darmstadt（达姆施塔特工业大学）
In this paper, we propose a convergent parallel best-response algorithm with the exact line search for the nondifferentiable nonconvex sparsity-regularized rank minimization problem. On the one hand, it exhibits a faster convergence than subgradient algorithms and block coordinate descent algorithms. On the other hand, its convergence to a stationary point is guaranteed, while ADMM algorithms only converge for convex problems. Furthermore, the exact line search procedure in the proposed algorithm is performed efficiently in closed-form to avoid the meticulous choice of stepsizes, which is howe…
In context of document classification, where in a corpus of documents their label tags are readily known, an opportunity lies in utilizing label information to learn document representation spaces with better discriminative properties. To this end, in this paper application of a Variational Bayesian Supervised Nonnegative Matrix Factorization (supervised vbNMF) with label-driven sparsity structure of coefficients is proposed for learning of discriminative nonsubtractive latent semantic components occuring in TF-IDF document representations. Constraints are such that the components pursued are …
Low-rank approximation is a common tool used to accelerate kernel methods: the $n \times n$ kernel matrix $K$ is approximated via a rank-$k$ matrix $\tilde K$ which can be stored in much less space and processed more quickly. In this work we study the limits of computationally efficient low-rank kernel approximation. We show that for a broad class of kernels, including the popular Gaussian and polynomial kernels, computing a relative error $k$-rank approximation to $K$ is at least as difficult as multiplying the input data matrix $A \in \mathbb{R}^{n \times d}$ by an arbitrary matrix $C \in \m…
We present a new algorithm for finding a near optimal low-rank approximation of a matrix $A$ in $O(nnz(A))$ time. Our method is based on a recursive sampling scheme for computing a representative subset of $A$'s columns, which is then used to find a low-rank approximation. This approach differs substantially from prior $O(nnz(A))$ time algorithms, which are all based on fast Johnson-Lindenstrauss random projections. It matches the guarantees of these methods while offering a number of advantages. Not only are sampling algorithms faster for sparse and structured data, but they can also be appli…
Pivoting Factorization: A Compact Meta Low-Rank Representation of Sparsity for Efficient Inference in Large Language Models
Center for Complex Network Intelligence (CCNI), Tsinghua Laboratory of Brain and Intelligence (THBI), Department of Psychological and Cognitive Sciences（复杂网络智能中心（CCNI）、清华脑智能实验室（THBI）、心理与认知科学系）；Department of Computer Science（计算机科学系）；Department of Biomedical Engineering, Tsinghua University, China（生物医学工程系，清华大学，中国）
The Hong Kong University of Science and Technology（香港科技大学）
In this paper, the estimation problem for sparse reduced rank regression (SRRR) model is considered. The SRRR model is widely used for dimension reduction and variable selection with applications in signal processing, econometrics, etc. The problem is formulated to minimize the least squares loss with a sparsity-inducing penalty considering an orthogonality constraint. Convex sparsity-inducing functions have been used for SRRR in literature. In this work, a nonconvex function is proposed for better sparsity inducing. An efficient algorithm is developed based on the alternating minimization (or…
The University of Tokyo（东京大学）；Austrian Academy of Sciences（奥地利科学院）；University of Cambridge（剑桥大学）；University of Warwick（华威大学）；RIKEN AIP（日本理化学研究所）
Kernel methods augmented with random features give scalable algorithms for learning from big data. But it has been computationally hard to sample random features according to a probability distribution that is optimized for the data, so as to minimize the required number of features for achieving the learning to a desired accuracy. Here, we develop a quantum algorithm for sampling from this optimized distribution over features, in runtime $O(D)$ that is linear in the dimension $D$ of the input data. Our algorithm achieves an exponential speedup in $D$ compared to any known classical algorithm …
Most existing approaches address multi-view subspace clustering problem by constructing the affinity matrix on each view separately and afterwards propose how to extend spectral clustering algorithm to handle multi-view data. This paper presents an approach to multi-view subspace clustering that learns a joint subspace representation by constructing affinity matrix shared among all views. Relying on the importance of both low-rank and sparsity constraints in the construction of the affinity matrix, we introduce the objective that balances between the agreement across different views, while at …
Sparsity-based approaches have been popular in many applications in image processing and imaging. Compressed sensing exploits the sparsity of images in a transform domain or dictionary to improve image recovery from undersampled measurements. In the context of inverse problems in dynamic imaging, recent research has demonstrated the promise of sparsity and low-rank techniques. For example, the patches of the underlying data are modeled as sparse in an adaptive dictionary domain, and the resulting image and dictionary estimation from undersampled measurements is called dictionary-blind compress…
The Scaling Law of Evaluation Failure: Why Simple Averaging Collapses Under Data Sparsity and Item Difficulty Gaps, and How Item Response Theory Recovers Ground Truth Across Domains
Exploring Sparsity for Parameter Efficient Fine Tuning Using Wavelets for Vision
Dept. of Electrical and Electronics Engineering, Koç University（电子工程系，科奇大学）；Codeway AI Research（Codeway人工智能研究）；Dept. of Computer Engineering, Middle East Technical University（计算机工程系，中东技术大学）
该研究提出小波微调（WaveFT），利用权重矩阵小波域的稀疏更新实现细粒度参数控制，在视觉任务的参数高效微调方法中达最优，已纳入Hugging Face PEFT库。
Sparse tensor best rank-1 approximation (BR1Approx), which is a sparsity generalization of the dense tensor BR1Approx, and is a higher-order extension of the sparse matrix BR1Approx, is one of the most important problems in sparse tensor decomposition and related problems arising from statistics and machine learning. By exploiting the multilinearity as well as the sparsity structure of the problem, four approximation algorithms are proposed, which are easily implemented, of low computational complexity, and can serve as initial procedures for iterative algorithms. In addition, theoretically gu…
Duke University, Fuqua School of Business（杜克大学富卡商学院）；Institute for Data, Systems, and Society, MIT（麻省理工学院数据、系统与社会研究所）；Operations Research Center, MIT（麻省理工学院运筹学中心）
We consider a seller offering a large network of $N$ products over a time horizon of $T$ periods. The seller does not know the parameters of the products' linear demand model, and can dynamically adjust product prices to learn the demand model based on sales observations. The seller aims to minimize its pseudo-regret, i.e., the expected revenue loss relative to a clairvoyant who knows the underlying demand model. We consider a sparse set of demand relationships between products to characterize various connectivity properties of the product network. In particular, we study three different spars…
Most learning methods with rank or sparsity constraints use convex relaxations, which lead to optimization with the nuclear norm or the $\ell_1$-norm. However, several important learning applications cannot benefit from this approach as they feature these convex norms as constraints in addition to the non-convex rank and sparsity constraints. In this setting, we derive efficient sparse projections onto the simplex and its extension, and illustrate how to use them to solve high-dimensional learning problems in quantum tomography, sparse density estimation and portfolio selection with non-convex…
The de-facto standard approach of promoting sparsity by means of $\ell_1$-regularization becomes ineffective in the presence of simplex constraints, i.e.,~the target is known to have non-negative entries summing up to a given constant. The situation is analogous for the use of nuclear norm regularization for low-rank recovery of Hermitian positive semidefinite matrices with given trace. In the present paper, we discuss several strategies to deal with this situation, from simple to more complex. As a starting point, we consider empirical risk minimization (ERM). It follows from existing theory …
Johns Hopkins University（约翰斯·霍普金斯大学）；Massachusetts Institute of Technology（麻省理工学院）；U.S. Army Research Laboratory（美国陆军研究实验室）
In this paper, we propose a general collaborative sparse representation framework for multi-sensor classification, which takes into account the correlations as well as complementary information between heterogeneous sensors simultaneously while considering joint sparsity within each sensor's observations. We also robustify our models to deal with the presence of sparse noise and low-rank interference signals. Specifically, we demonstrate that incorporating the noise or interference signal as a low-rank component in our models is essential in a multi-sensor classification problem when multiple …
Max-Planck-Institut für Informatik（马克斯·普朗克信息学研究所）；Carnegie Mellon University（卡内基梅隆大学）
We propose an input sparsity time sampling algorithm that can spectrally approximate the Gram matrix corresponding to the $q$-fold column-wise tensor product of $q$ matrices using a nearly optimal number of samples, improving upon all previously known methods by poly$(q)$ factors. Furthermore, for the important special case of the $q$-fold self-tensoring of a dataset, which is the feature matrix of the degree-$q$ polynomial kernel, the leading term of our method's runtime is proportional to the size of the input dataset and has no dependence on $q$. Previous techniques either incur poly$(q)$ s…
Developing shopping experiences that delight the customer requires businesses to understand customer taste. This work reports a method to learn the shopping preferences of frequent shoppers to an online gift store by combining ideas from retail analytics and statistical learning with sparsity. Shopping activity is represented as a bipartite graph. This graph is refined by applying sparsity-based statistical learning methods. These methods are interpretable and reveal insights about customers' preferences as well as products driving revenue to the store.
École Polytechnique Fédérale de Lausanne（洛桑联邦理工学院）；Swisscom（瑞士电信）
In this paper, we describe a method to tackle data sparsity and create recommendations in domains with limited knowledge about user preferences. We expand the variational autoencoder collaborative filtering from a single-domain to a multi-domain setting. The intuition is that user-item interactions in a source domain can augment the recommendation quality in a target domain. The intuition can be taken to its extreme, where, in a cross-domain setup, the user history in a source domain is enough to generate high-quality recommendations in a target one. We thus create a Product-of-Experts (POE) a…
Michigan State University（密歇根州立大学）；Korea Advanced Institute of Science & Technology（韩国科学技术院）；University of Michigan（密歇根大学）
The field of medical image reconstruction has seen roughly four types of methods. The first type tended to be analytical methods, such as filtered back-projection (FBP) for X-ray computed tomography (CT) and the inverse Fourier transform for magnetic resonance imaging (MRI), based on simple mathematical models for the imaging systems. These methods are typically fast, but have suboptimal properties such as poor resolution-noise trade-off for CT. A second type is iterative reconstruction methods based on more complete models for the imaging system physics and, where appropriate, models for the …
Northeast Normal University（东北师范大学）；University of Washington（华盛顿大学）；Carnegie Mellon University（卡内基梅隆大学）
In the total least squares problem, one is given an $m \times n$ matrix $A$, and an $m \times d$ matrix $B$, and one seeks to "correct" both $A$ and $B$, obtaining matrices $\hat{A}$ and $\hat{B}$, so that there exists an $X$ satisfying the equation $\hat{A}X = \hat{B}$. Typically the problem is overconstrained, meaning that $m \gg \max(n,d)$. The cost of the solution $\hat{A}, \hat{B}$ is given by $\|A-\hat{A}\|_F^2 + \|B - \hat{B}\|_F^2$. We give an algorithm for finding a solution $X$ to the linear system $\hat{A}X=\hat{B}$ for which the cost $\|A-\hat{A}\|_F^2 + \|B-\hat{B}\|_F^2$ is at mo…
Center for Security Theory and Algorithmic Research（安全理论与算法研究中心）；International Institute of Information Technology Hyderabad（海得拉巴国际信息技术学院）
Sparse neural networks are shown to give accurate predictions competitive to denser versions, while also minimizing the number of arithmetic operations performed. However current hardware like GPU's can only exploit structured sparsity patterns for better efficiency. Hence the run time of a sparse neural network may not correspond to the arithmetic operations required. In this work, we propose RBGP( Ramanujan Bipartite Graph Product) framework for generating structured multi level block sparse neural networks by using the theory of Graph products. We also propose to use products of Ramanujan g…
We analyze the decomposition of a data matrix, assumed to be a superposition of a low-rank component and a component which is sparse in a known dictionary, using a convex demixing method. We provide a unified analysis, encompassing both undercomplete and overcomplete dictionary cases, and show that the constituent components can be successfully recovered under some relatively mild assumptions up to a certain $\textit{global}$ sparsity level. Further, we corroborate our theoretical results by presenting empirical evaluations in terms of phase transitions in rank and sparsity for various diction…
University of Minnesota（明尼苏达大学）；Princeton University（普林斯顿大学）
We consider the decomposition of a data matrix assumed to be a superposition of a low-rank matrix and a component which is sparse in a known dictionary, using a convex demixing method. We consider two sparsity structures for the sparse factor of the dictionary sparse component, namely entry-wise and column-wise sparsity, and provide a unified analysis, encompassing both undercomplete and the overcomplete dictionary cases, to show that the constituent matrices can be successfully recovered under some relatively mild conditions on incoherence, sparsity, and rank. We leverage these results to loc…
Search is a prominent channel for discovering products on an e-commerce platform. Ranking products retrieved from search becomes crucial to address customer's need and optimize for business metrics. While learning to Rank (LETOR) models have been extensively studied and have demonstrated efficacy in the context of web search; it is a relatively new research area to be explored in the e-commerce. In this paper, we present a framework for building LETOR model for an e-commerce platform. We analyze user queries and propose a mechanism to segment queries between broad and narrow based on user's in…
