Related papers: Matrix best approximation in the spectral norm
The problem of computing spectra of operators is arguably one of the most investigated areas of computational mathematics. However, the problem of computing spectra of general bounded infinite matrices has only recently been solved. We…
This paper studies two structured approximation problems: (1) Recovering a corrupted low-rank Toeplitz matrix and (2) recovering the range of a Fourier matrix from a single observation. Both problems are computationally challenging because…
In this paper, we introduce the Maximum Matrix Contraction problem, where we aim to contract as much as possible a binary matrix in order to maximize its density. We study the complexity and the polynomial approximability of the problem.…
This paper studies the matrix completion problem under arbitrary sampling schemes. We propose a new estimator incorporating both max-norm and nuclear-norm regularization, based on which we can conduct efficient low-rank matrix recovery…
Mobile gamma-ray spectrometry (MGRS) systems are essential for localizing, identifying, and quantifying gamma-ray sources in complex environments. Full-spectrum template matching offers the highest accuracy and sensitivity for these tasks…
It has been observed that the performances of many high-dimensional estimation problems are universal with respect to underlying sensing (or design) matrices. Specifically, matrices with markedly different constructions seem to achieve…
We study the problem of selecting a subset of k random variables from a large set, in order to obtain the best linear prediction of another variable of interest. This problem can be viewed in the context of both feature selection and sparse…
We provide an asymptotically tight, computationally efficient approximation of the joint spectral radius of a set of matrices using sum of squares (SOS) programming. The approach is based on a search for an SOS polynomial that proves…
We revisit a natural variant of geometric set cover, called minimum-membership geometric set cover (MMGSC). In this problem, the input consists of a set $S$ of points and a set $\mathcal{R}$ of geometric objects, and the goal is to find a…
Deep Matrix Factorization (DMF) is an emerging approach to the problem of matrix completion. Recent works have established that gradient descent applied to a DMF model induces an implicit regularization on the rank of the recovered matrix.…
In this paper, we consider the problem of approximating a given matrix with a matrix whose eigenvalues lie in some specific region \Omega, within the complex plane. More precisely, we consider three types of regions and their intersections:…
Biclustering techniques have been widely used to identify homogeneous subgroups within large data matrices, such as subsets of genes similarly expressed across subsets of patients. Mining a max-sum sub-matrix is a related but distinct…
In this paper, we consider optimal low-rank regularized inverse matrix approximations and their applications to inverse problems. We give an explicit solution to a generalized rank-constrained regularized inverse approximation problem,…
Over the last decade, an approach that has gained a lot of popularity to tackle nonparametric testing problems on general (i.e., non-Euclidean) domains is based on the notion of reproducing kernel Hilbert space (RKHS) embedding of…
We extend the result on the spectral projected gradient method by Birgin et al. in 2000 to a log-determinant semidefinite problem (SDP) with linear constraints and propose a spectral projected gradient method for the dual problem. Our…
Randomized matrix sparsification has proven to be a fruitful technique for producing faster algorithms in applications ranging from graph partitioning to semidefinite programming. In the decade or so of research into this technique, the…
We revisit the matrix problems sparse null space and matrix sparsification, and show that they are equivalent. We then proceed to seek algorithms for these problems: We prove the hardness of approximation of these problems, and also give a…
Graph is a highly generic and diverse representation, suitable for almost any data processing problem. Spectral graph theory has been shown to provide powerful algorithms, backed by solid linear algebra theory. It thus can be extremely…
Given a set of points in the plane, the \textsc{General Position Subset Selection} problem is that of finding a maximum-size subset of points in general position, i.e., with no three points collinear. The problem is known to be ${\rm…
This paper describes a suite of algorithms for constructing low-rank approximations of an input matrix from a random linear image of the matrix, called a sketch. These methods can preserve structural properties of the input matrix, such as…