Tridiagonal Maximum-Entropy Sampling and Tridiagonal Masks

TL;DR

This paper develops efficient algorithms and bounds for the maximum-entropy sampling problem (MESP) when the covariance matrix or its inverse is tridiagonal or has a spider graph structure, including a local-search method for tridiagonal masks.

Contribution

It introduces a dynamic-programming algorithm for MESP with tridiagonal or spider graph covariance matrices and analyzes tridiagonal masks for fast upper bounds.

Findings

Efficient dynamic programming for tridiagonal covariance matrices.

A class of arrowhead matrices solvable by greedy algorithms.

A local-search method for tridiagonal masks that leverages fast computations.

Abstract

The NP-hard maximum-entropy sampling problem (MESP) seeks a maximum (log-)determinant principal submatrix, of a given order, from an input covariance matrix $C$. We give an efficient dynamic-programming algorithm for MESP when $C$ (or its inverse) is tridiagonal and generalize it to the situation where the support graph of $C$ (or its inverse) is a spider graph with a constant number of legs (and beyond). We give a class of arrowhead covariance matrices $C$ for which a natural greedy algorithm solves MESP. A \emph{mask} $M$ for MESP is a correlation matrix with which we pre-process $C$, by taking the Hadamard product $M\circ C$. Upper bounds on MESP with $M\circ C$ give upper bounds on MESP with $C$. Most upper-bounding methods are much faster to apply, when the input matrix is tridiagonal, so we consider tridiagonal masks $M$ (which yield tridiagonal $M\circ C$). We make a detailed analysis of such tridiagonal masks, and develop a combinatorial local-search based upper-bounding method that takes advantage of fast computations on tridiagonal matrices.