A Local Inverse Formula and a Factorization

TL;DR

This paper discusses a local inverse formula for matrices with specific sparsity patterns, enabling efficient computation of inverses and revealing connections to various mathematical and machine learning applications.

Contribution

It introduces a generalized local inverse formula for matrices with chordal sparsity patterns and explains its interpretation as a matrix factorization.

Findings

The local inverse formula applies to matrices with chordal sparsity patterns.

The formula enables efficient inverse computation using local information.

Connections to maximum entropy, wavelets, and graphical models are established.

Abstract

When a matrix has a banded inverse there is a remarkable formula that quickly computes that inverse, using only local information in the original matrix. This local inverse formula holds more generally, for matrices with sparsity patterns that are examples of chordal graphs or perfect eliminators. The formula has a long history going back at least as far as the completion problem for covariance matrices with missing data. Maximum entropy estimates, log-determinants, rank conditions, the Nullity Theorem and wavelets are all closely related, and the formula has found wide applications in machine learning and graphical models. We describe that local inverse and explain how it can be understood as a matrix factorization.