arXiv research

Rank-one measurements limit feasible sets for low-rank PSD matrices.

This paper tackles fitting multilevel low rank matrices by addressing three problems.

New algorithm learns low-rank matrices with linear number of samples.

Algorithm recovers multiple low-rank matrices from unlabeled data.

Matrices of (approximate) low rank are pervasive in data science, appearing in recommender systems, movie preferences, topic models, medical records, and genomics. While there is a vast literature on how to exploit low rank structure in these datasets, there is less attention on explaining why the low rank structure ap…

New framework solves low-rank optimization problems to certifiable optimality.

Develops PRPCA for smooth image recovery combining low-rank and smoothness.

We address some theoretical guarantees for Schatten-$p$ quasi-norm minimization ($p \in (0,1]$) in recovering low-rank matrices from compressed linear measurements. Firstly, using null space properties of the measurement operator, we provide a sufficient condition for exact recovery of low-rank matrices. This condition…

New method reduces computational cost for nonnegative low rank matrix approximation.

Weight Decay induces low-rank weight matrices in neural networks, improving generalization.

LOCUS separates brain network connectivity matrices efficiently.

Sparse PCA is a widely used technique for high-dimensional data analysis. In this paper, we propose a new method called low-rank principal eigenmatrix analysis. Different from sparse PCA, the dominant eigenvectors are allowed to be dense but are assumed to have a low-rank structure when matricized appropriately. Such a…

We develop latent variable models for Bayesian learning based low-rank matrix completion and reconstruction from linear measurements. For under-determined systems, the developed methods are shown to reconstruct low-rank matrices when neither the rank nor the noise power is known a-priori. We derive relations between th…

Robust PCA is a widely used statistical procedure to recover a underlying low-rank matrix with grossly corrupted observations. This work considers the problem of robust PCA as a nonconvex optimization problem on the manifold of low-rank matrices, and proposes two algorithms (for two versions of retractions) based on ma…

CMF is a technique for simultaneously learning low-rank representations based on a collection of matrices with shared entities. A typical example is the joint modeling of user-item, item-property, and user-feature matrices in a recommender system. The key idea in CMF is that the embeddings are shared across the matrice…

Low-rank approximations of data matrices are an important dimensionality reduction tool in machine learning and regression analysis. We consider the case of categorical variables, where it can be formulated as the problem of finding low-rank approximations to Boolean matrices. In this paper we give what is to the best …

Matrix completion is a problem that arises in many data-analysis settings where the input consists of a partially-observed matrix (e.g., recommender systems, traffic matrix analysis etc.). Classical approaches to matrix completion assume that the input partially-observed matrix is low rank. The success of these methods…

New method corrects quantization errors in LLMs using low-rank matrices.

TensorGuide improves LoRA efficiency and expressivity through joint tensor-train optimization.

Given the superposition of a low-rank matrix plus the product of a known fat compression matrix times a sparse matrix, the goal of this paper is to establish deterministic conditions under which exact recovery of the low-rank and sparse components becomes possible. This fundamental identifiability issue arises with tra…

Multiresolution Matrix Factorization (MMF) was recently introduced as an alternative to the dominant low-rank paradigm in order to capture structure in matrices at multiple different scales. Using ideas from multiresolution analysis (MRA), MMF teased out hierarchical structure in symmetric matrices by constructing a se…

This paper uncovers the low-rank structure of neural network Hessians.

New algorithms improve RPCA for large matrices with upper rank bounds.

Mirror descent algorithm recovers low-rank matrices in matrix sensing.

We address the problem of minimizing a convex function over the space of large matrices with low rank. While this optimization problem is hard in general, we propose an efficient greedy algorithm and derive its formal approximation guarantees. Each iteration of the algorithm involves (approximately) finding the left an…

Efficient algorithm for Hadamard decomposition of matrices.

Paper finds a lower bound for estimating low-rank matrices in logistic regression.

New method improves matrix completion accuracy, especially in noisy data.

In this letter, we propose an algorithm for recovery of sparse and low rank components of matrices using an iterative method with adaptive thresholding. In each iteration, the low rank and sparse components are obtained using a thresholding operator. This algorithm is fast and can be implemented easily. We compare it w…

ScaledGD improves gradient descent for ill-conditioned low-rank matrix estimation.

Riemannian gradient descent escapes some spurious critical points on low-rank matrix manifold.

Paper proposes a new algorithm for graph learning with covariance constraints.

Study evaluates thresholds for removing noise from DNN weights using random matrix theory.

Density matrices are positively semi-definite Hermitian matrices with unit trace that describe the states of quantum systems. Many quantum systems of physical interest can be represented as high-dimensional low rank density matrices. A popular problem in {\it quantum state tomography} (QST) is to estimate the unknown l…

Paper proposes a new method to separate low rank and sparse matrices without bias.

A new method reduces high-dimensional filtering to quadratic complexity.

Algorithm compresses large matrices by approximating them as low rank and low precision factors.

Efficiently implements MEG for low-rank matrix optimization problems.

The density matrices are positively semi-definite Hermitian matrices of unit trace that describe the state of a quantum system. The goal of the paper is to develop minimax lower bounds on error rates of estimation of low rank density matrices in trace regression models used in quantum state tomography (in particular, i…

In the paper, we consider the problem of link prediction in time-evolving graphs. We assume that certain graph features, such as the node degree, follow a vector autoregressive (VAR) model and we propose to use this information to improve the accuracy of prediction. Our strategy involves a joint optimization procedure …

Geometric families of low-rank covariances improve flexibility and tractability in high dimensions.

Flora uses random projections to achieve high-rank updates with low memory usage.

Estimation of low-rank matrices is of significant interest in a range of contemporary applications. In this paper, we introduce a rank-one projection model for low-rank matrix recovery and propose a constrained nuclear norm minimization method for stable recovery of low-rank matrices in the noisy case. The procedure is…

Robust principal component analysis (RPCA) has drawn significant attentions due to its powerful capability in recovering low-rank matrices as well as successful appplications in various real world problems. The current state-of-the-art algorithms usually need to solve singular value decomposition of large matrices, whi…

Unified model for tensor completion using low-rank and sparse Tucker decomposition.

We address the collective matrix completion problem of jointly recovering a collection of matrices with shared structure from partial (and potentially noisy) observations. To ensure well--posedness of the problem, we impose a joint low rank structure, wherein each component matrix is low rank and the latent space of th…

Improves BBVI for high-dimensional Gaussian approximations by using low-rank approximations.

We analyze a class of estimators based on convex relaxation for solving high-dimensional matrix decomposition problems. The observations are noisy realizations of a linear transformation $\mathfrak{X}$ of the sum of an approximately) low rank matrix $Θ^\star$ with a second matrix $Γ^\star$ endowed with a complementary …