Generalized Interlacing Families: New Error Bounds for CUR Matrix Decompositions

TL;DR

This paper extends interlacing polynomial methods to handle polynomials of varying degrees, providing new error bounds for CUR matrix decompositions and row subset selection, along with a deterministic algorithm for classical CUR problems.

Contribution

It introduces generalized interlacing families of polynomials, deriving tighter spectral norm bounds for CUR decompositions and establishing the first bounds for row subset selection.

Findings

Tighter spectral norm error bounds for classical CUR matrix decompositions.

First known spectral norm error bounds for row subset selection.

A deterministic polynomial-time algorithm for classical CUR problems.

Abstract

This paper introduces the concept of generalized interlacing families of polynomials, which extends the classical interlacing polynomial method to handle polynomials of varying degrees. We establish a fundamental property for these families, proving the existence of a polynomial with a desired degree whose smallest root is greater than or equal to the smallest root of the expected polynomial. Applying this framework to the generalized CUR matrix approximation problem, we derive a theoretical upper bound on the spectral norm of a residual matrix, expressed in terms of the largest root of the expected polynomial. We then explore two important special cases: the classical CUR matrix decompositions and the row subset selection problem. For classical CUR matrix decompositions, we derive an explicit upper bound for the largest root of the expected polynomial. This yields a tighter spectral norm error bound for the residual matrix compared to many existing results. Furthermore, we present a deterministic polynomial-time algorithm for solving the classical CUR problem under certain matrix conditions. For the row subset selection problem, we establish the first known spectral norm error bound. This paper extends the applicability of interlacing families and deepens the theoretical foundations of CUR matrix decompositions and related approximation problems.