Related papers: Matrix-Free Approximate Equilibration
Tensor completion is a natural higher-order generalization of matrix completion where the goal is to recover a low-rank tensor from sparse observations of its entries. Existing algorithms are either heuristic without provable guarantees,…
This paper introduces the Nystr\"om PCG algorithm for solving a symmetric positive-definite linear system. The algorithm applies the randomized Nystr\"om method to form a low-rank approximation of the matrix, which leads to an efficient…
Scaling problems have a rich and diverse history, and thereby have found numerous applications in several fields of science and engineering. For instance, the matrix scaling problem has had applications ranging from theoretical computer…
Suppose we are given a matrix that is formed by adding an unknown sparse matrix to an unknown low-rank matrix. Our goal is to decompose the given matrix into its sparse and low-rank components. Such a problem arises in a number of…
Matrices can be augmented by adding additional columns such that a partitioning of the matrix in blocks of rows defines mutually orthogonal subspaces. This augmented system can then be solved efficiently by a sum of projections onto these…
Low-rank modeling has many important applications in computer vision and machine learning. While the matrix rank is often approximated by the convex nuclear norm, the use of nonconvex low-rank regularizers has demonstrated better empirical…
Estimates of the approximate factor model are increasingly used in empirical work. Their theoretical properties, studied some twenty years ago, also laid the ground work for analysis on large dimensional panel data models with cross-section…
The process of alternately row scaling and column scaling a positive $n \times n$ matrix $A$ converges to a doubly stochastic positive $n \times n$ matrix $S(A)$, often called the \emph{Sinkhorn limit} of $A$. The main result in this paper…
In this era of large-scale data, distributed systems built on top of clusters of commodity hardware provide cheap and reliable storage and scalable processing of massive data. Here, we review recent work on developing and implementing…
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…
The convergence rates of iterative methods for solving a linear system $\mathbf{A} x = b$ typically depend on the condition number of the matrix $\mathbf{A}$. Preconditioning is a common way of speeding up these methods by reducing that…
In graph theory a partition of the vertex set of a graph is called equitable if for all pairs of cells all vertices in one cell have an equal number of neighbours in the other cell. Considering the implications for the adjacency matrix one…
We examine the approach to equilibrium of the micromaser. Analytic methods are first used to show that for large times (i.e. many atoms) the convergence is governed by the next to leading eigenvalue of the corresponding discrete evolution…
We revisit Matrix Balancing, a pre-conditioning task used ubiquitously for computing eigenvalues and matrix exponentials. Since 1960, Osborne's algorithm has been the practitioners' algorithm of choice and is now implemented in most…
Factorization-based gradient descent is a scalable and efficient algorithm for solving low-rank matrix completion. Recent progress in structured non-convex optimization has offered global convergence guarantees for gradient descent under…
In many applications, e.g., recommender systems and traffic monitoring, the data comes in the form of a matrix that is only partially observed and low rank. A fundamental data-analysis task for these datasets is matrix completion, where the…
We consider the problem of finding the optimal diagonal preconditioner for a positive definite matrix. Although this problem has been shown to be solvable and various methods have been proposed, none of the existing approaches are scalable…
Matrix completion refers to completing a low-rank matrix from a few observed elements of its entries and has been known as one of the significant and widely-used problems in recent years. The required number of observations for exact…
Several methods for computing the action of the matrix exponential $\mathrm{e}^{\boldsymbol{A}} \boldsymbol{b}$ are expressed by substituting $\boldsymbol{A}$ into a rational approximation of the scalar exponential function. The error of…
Originally developed for imputing missing entries in low rank, or approximately low rank matrices, matrix completion has proven widely effective in many problems where there is no reason to assume low-dimensional linear structure in the…