Showing papers by "Joel A. Tropp published in 2019"

Streaming Low-Rank Matrix Approximation With An Application To Scientific Simulation

Abstract: This paper argues that randomized linear sketching is a natural tool for on-the-fly compression of data matrices that arise from large-scale scientific simulations and data collection. The technica...

Scalable Semidefinite Programming

Abstract: Semidefinite programming (SDP) is a powerful framework from convex optimization that has striking potential for data science applications. This paper develops a provably correct randomized algorithm for solving large, weakly constrained SDP problems by economizing on the storage and arithmetic costs. Numerical evidence shows that the method is effective for a range of applications, including relaxations of MaxCut, abstract phase retrieval, and quadratic assignment. Running on a laptop equivalent, the algorithm can handle SDP instances where the matrix variable has over $10^{14}$ entries.

An Optimal-Storage Approach to Semidefinite Programming using Approximate Complementarity

Abstract: This paper develops a new storage-optimal algorithm that provably solves generic semidefinite programs (SDPs) in standard form. This method is particularly effective for weakly constrained SDPs. The key idea is to formulate an approximate complementarity principle: Given an approximate solution to the dual SDP, the primal SDP has an approximate solution whose range is contained in the eigenspace with small eigenvalues of the dual slack matrix. For weakly constrained SDPs, this eigenspace has very low dimension, so this observation significantly reduces the search space for the primal solution. This result suggests an algorithmic strategy that can be implemented with minimal storage: (1) Solve the dual SDP approximately; (2) compress the primal SDP to the eigenspace with small eigenvalues of the dual slack matrix; (3) solve the compressed primal SDP. The paper also provides numerical experiments showing that this approach is successful for a range of interesting large-scale SDPs.

Low-Rank Tucker Approximation of a Tensor From Streaming Data

Abstract: This paper describes a new algorithm for computing a low-Tucker-rank approximation of a tensor. The method applies a randomized linear map to the tensor to obtain a sketch that captures the important directions within each mode, as well as the interactions among the modes. The sketch can be extracted from streaming or distributed data or with a single pass over the tensor, and it uses storage proportional to the degrees of freedom in the output Tucker approximation. The algorithm does not require a second pass over the tensor, although it can exploit another view to compute a superior approximation. The paper provides a rigorous theoretical guarantee on the approximation error. Extensive numerical experiments show that that the algorithm produces useful results that improve on the state of the art for streaming Tucker decomposition.

Streaming Low-Rank Matrix Approximation with an Application to Scientific Simulation

Abstract: This paper argues that randomized linear sketching is a natural tool for on-the-fly compression of data matrices that arise from large-scale scientific simulations and data collection. The technical contribution consists in a new algorithm for constructing an accurate low-rank approximation of a matrix from streaming data. This method is accompanied by an a priori analysis that allows the user to set algorithm parameters with confidence and an a posteriori error estimator that allows the user to validate the quality of the reconstructed matrix. In comparison to previous techniques, the new method achieves smaller relative approximation errors and is less sensitive to parameter choices. As concrete applications, the paper outlines how the algorithm can be used to compress a Navier--Stokes simulation and a sea surface temperature dataset.

Binary component decomposition Part II: The asymmetric case.

Abstract: This paper studies the problem of decomposing a low-rank matrix into a factor with binary entries, either from {±1} or from {0,1}, and an unconstrained factor. The research answers fundamental questions about the existence and uniqueness of these decompositions. It also leads to tractable factorization algorithms that succeed under a mild deterministic condition. This work builds on a companion paper that addresses the related problem of decomposing a low-rank positive-semidefinite matrix into symmetric binary factors.

Matrix Concentration & Computational Linear Algebra

Abstract: These lecture notes were written to support the short course Matrix Concentration & Computational Linear Algebra delivered by the author at Ecole Normale Superieure in Paris from 1–5 July 2019 as part of the summer school “High-dimensional probability and algorithms.” The aim of this course is to present some practical computational applications of matrix concentration.

Binary Component Decomposition Part I: The Positive-Semidefinite Case

Abstract: This paper studies the problem of decomposing a low-rank positive-semidefinite matrix into symmetric factors with binary entries, either $\{\pm 1\}$ or $\{0,1\}$. This research answers fundamental questions about the existence and uniqueness of these decompositions. It also leads to tractable factorization algorithms that succeed under a mild deterministic condition. A companion paper addresses the related problem of decomposing a low-rank rectangular matrix into a binary factor and an unconstrained factor.