Rectangular maximum-volume submatrices and their applications

TL;DR

This paper defines and explores the concept of maximum-volume submatrices for rectangular matrices, extending previous work on square matrices, and demonstrates their applications in areas like recommender systems and linear system preconditioning.

Contribution

It introduces a new definition of volume for rectangular matrices, generalizes maximum-volume submatrix results, and links these to experimental design and practical applications.

Findings

Connection between rectangular volume and optimal experimental design

Estimates for coefficient growth and approximation error

Applications in recommender systems and linear system preconditioning

Abstract

We introduce a definition of the volume for a general rectangular matrix, which for square matrices is equivalent to the absolute value of the determinant. We generalize results for square maximum-volume submatrices to the case of rectangular maximal-volume submatrices, show connection of the rectangular volume with optimal experimental design and provide estimates for the growth of the coefficients and approximation error in spectral and Chebyshev norms. Three promising applications of such submatrices are presented: recommender systems, finding maximal elements in low-rank matrices and preconditioning of overdetermined linear systems. The code is available online.