Logarithmically Sparse Symmetric Matrices

TL;DR

This paper explores the properties of logarithmically sparse positive definite matrices, providing algebraic geometry tools to understand their structure, including formulas for dimension, degree bounds, and algorithms for implicitization.

Contribution

It introduces a novel algebraic geometric framework for analyzing logarithmically sparse matrices, including formulas, bounds, and algorithms for their characterization.

Findings

Derived a formula for the dimension of the Zariski closure.

Provided degree bounds for the associated algebraic variety.

Developed algorithms for implicitization and finding defining equations.

Abstract

A positive definite matrix is called logarithmically sparse if its matrix logarithm has many zero entries. Such matrices play a significant role in high-dimensional statistics and semidefinite optimization. In this paper, logarithmically sparse matrices are studied from the point of view of computational algebraic geometry: we present a formula for the dimension of the Zariski closure of a set of matrices with a given logarithmic sparsity pattern, give a degree bound for this variety and develop implicitization algorithms that allow to find its defining equations. We illustrate our approach with numerous examples.