Related papers: Fast Derandomized Low-rank Approximation and Exten…
We consider the problem of approximating an affinely structured matrix, for example a Hankel matrix, by a low-rank matrix with the same structure. This problem occurs in system identification, signal processing and computer algebra, among…
This survey explores modern approaches for computing low-rank approximations of high-dimensional matrices by means of the randomized SVD, randomized subspace iteration, and randomized block Krylov iteration. The paper compares the…
Low rank matrix recovery problems, including matrix completion and matrix sensing, appear in a broad range of applications. In this work we present GNMR -- an extremely simple iterative algorithm for low rank matrix recovery, based on a…
Many algorithms in scientific computing and data science take advantage of low-rank approximation of matrices and kernels, and understanding why nearly-low-rank structure occurs is essential for their analysis and further development. This…
We consider the problem of learning a low-rank matrix, constrained to lie in a linear subspace, and introduce a novel factorization for modeling such matrices. A salient feature of the proposed factorization scheme is it decouples the…
We consider the problem of recovering an unknown low-rank matrix X with (possibly) non-orthogonal, effectively sparse rank-1 decomposition from measurements y gathered in a linear measurement process A. We propose a variational formulation…
Alternating Minimization is a widely used and empirically successful heuristic for matrix completion and related low-rank optimization problems. Theoretical guarantees for Alternating Minimization have been hard to come by and are still…
We propose a general error analysis related to the low-rank approximation of a given real matrix in both the spectral and Frobenius norms. First, we derive deterministic error bounds that hold with some minimal assumptions. Second, we…
Low-rank tensor approximation techniques attempt to mitigate the overwhelming complexity of linear algebra tasks arising from high-dimensional applications. In this work, we study the low-rank approximability of solutions to linear systems…
Structural matrix-variate observations routinely arise in diverse fields such as multi-layer network analysis and brain image clustering. While data of this type have been extensively investigated with fruitful outcomes being delivered, the…
Low-rank approximation with zeros aims to find a matrix of fixed rank and with a fixed zero pattern that minimizes the Euclidean distance to a given data matrix. We study the critical points of this optimization problem using algebraic…
We study the problem of estimating low-rank matrices from linear measurements (a.k.a., matrix sensing) through nonconvex optimization. We propose an efficient stochastic variance reduced gradient descent algorithm to solve a nonconvex…
Low-rank matrix completion consists of computing a matrix of minimal complexity that recovers a given set of observations as accurately as possible. Unfortunately, existing methods for matrix completion are heuristics that, while highly…
In this paper, we propose a low rank approximation method for efficiently solving stochastic partial differential equations. Specifically, our method utilizes a novel low rank approximation of the stiffness matrices, which can significantly…
1. A standard Gaussian random matrix has full rank with probability 1 and is well-conditioned with a probability quite close to 1 and converging to 1 fast as the matrix deviates from square shape and becomes more rectangular. 2. If we…
This paper is concerned with the problem of low rank plus sparse matrix decomposition for big data. Conventional algorithms for matrix decomposition use the entire data to extract the low-rank and sparse components, and are based on…
We study the problem of finding structured low-rank matrices using nuclear norm regularization where the structure is encoded by a linear map. In contrast to most known approaches for linearly structured rank minimization, we do not (a) use…
Low rank inference on matrices is widely conducted by optimizing a cost function augmented with a penalty proportional to the nuclear norm $\Vert \cdot \Vert_*$. However, despite the assortment of computational methods for such problems,…
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…
The purpose of this text is to provide an accessible introduction to a set of recently developed algorithms for factorizing matrices. These new algorithms attain high practical speed by reducing the dimensionality of intermediate…