Related papers: Disk matrices and the proximal mapping for the num…
We study the following optimization problem over a dynamical system that consists of several linear subsystems: Given a finite set of $n\times n$ matrices and an $n$-dimensional vector, find a sequence of $K$ matrices, each chosen from the…
We propose an extremely versatile approach to address a large family of matrix nearness problems, possibly with additional linear constraints. Our method is based on splitting a matrix nearness problem into two nested optimization problems,…
We consider the densest submatrix problem, which seeks the submatrix of fixed size of a given binary matrix that contains the most nonzero entries. This problem is a natural generalization of fundamental problems in combinatorial…
We consider the problem of computing the data-cube marginals of a fixed order $k$ (i.e., all marginals that aggregate over $k$ dimensions), using a single round of MapReduce. The focus is on the relationship between the reducer size (number…
New algorithms are presented for numerical conformal mapping based on rational approximations and the solution of Dirichlet problems by least-squares fitting on the boundary. The methods are targeted at regions with corners, where the…
We solve the problem of best approximation by partial isometries of given rank to an arbitrary rectangular matrix, when the distance is measured in any unitarily invariant norm. In the case where the norm is strictly convex, we parametrize…
We study the problem of approximating the cone of positive semidefinite (PSD) matrices with a cone that can be described by smaller-sized PSD constraints. Specifically, we ask the question: "how closely can we approximate the set of…
We display four approximation theorems for manifold-valued mappings. The first one approximates holomorphic embeddings on pseudoconvex domains in $\Bbb C^n$ with holomorphic embeddings with dense images. The second theorem approximates…
Randomized sampling has recently been demonstrated to be an efficient technique for computing approximate low-rank factorizations of matrices for which fast methods for computing matrix vector products are available. This paper describes an…
We develop tractable convex relaxations for rank-constrained quadratic optimization problems over $n \times m$ matrices, a setting for which tractable relaxations are typically only available when the objective or constraints admit spectral…
This short course offers a new perspective on randomized algorithms for matrix computations. It explores the distinct ways in which probability can be used to design algorithms for numerical linear algebra. Each design template is…
We define geometric matrix midranges for positive definite Hermitian matrices and study the midrange problem from a number of perspectives. Special attention is given to the midrange of two positive definite matrices before considering the…
Matrices arising in scientific applications frequently admit linear low-rank approximations due to smoothness in the physical and/or temporal domain of the problem. In large-scale problems, computing an optimal low-rank approximation can be…
We address composite optimization problems, which consist in minimizing the sum of a smooth and a merely lower semicontinuous function, without any convexity assumptions. Numerical solutions of these problems can be obtained by proximal…
Matrix multiplication is a fundamental computation in many scientific disciplines. In this paper, we show that novel fast matrix multiplication algorithms can significantly outperform vendor implementations of the classical algorithm and…
Consider the collection of all binary matrices having a specific sequence of row and column sums and consider sampling binary matrices uniformly from this collection. Practical algorithms for exact uniform sampling are not known, but there…
The covariance matrix of measurements of Markov random fields (processes) has useful properties that allow to develop effective computational algorithms for many problems in the study of Markov fields on the basis of field observations…
Positive semi-definite matrices commonly occur as normal matrices of least squares problems in statistics or as kernel matrices in machine learning and approximation theory. They are typically large and dense. Thus algorithms to solve…
A collection of algorithms is described for numerically computing with smooth functions defined on the unit disk. Low rank approximations to functions in polar geometries are formed by synthesizing the disk analogue of the double Fourier…
Convex nonsmooth optimization problems, whose solutions live in very high dimensional spaces, have become ubiquitous. To solve them, the class of first-order algorithms known as proximal splitting algorithms is particularly adequate: they…