A Local Inverse Formula and a Factorization
This is a theoretical contribution that unifies and clarifies a known formula for researchers in matrix theory and graphical models.
The paper presents a local inverse formula for matrices with banded or chordal sparsity patterns, tracing its history and applications in machine learning and graphical models, and explains it as a matrix factorization.
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.