An Iterative Block Matrix Inversion (IBMI) Algorithm for Symmetric Positive Definite Matrices with Applications to Covariance Matrices

TL;DR

This paper introduces an iterative block matrix inversion algorithm (IBMI) for efficiently approximating the inverse of large symmetric positive definite matrices, with proven convergence and practical applications to covariance matrices.

Contribution

The paper proposes a novel IBMI algorithm that uses block matrix inversion and demonstrates its convergence for large symmetric positive definite matrices.

Findings

Two-block approach converges for any positive definite matrix

Multi-block overlapping approach also converges

Numerical results validate the effectiveness of the algorithm

Abstract

Obtaining the inverse of a large symmetric positive definite matrix $\mathcal{A}\in\mathbb{R}^{p\times p}$ is a continual challenge across many mathematical disciplines. The computational complexity associated with direct methods can be prohibitively expensive, making it infeasible to compute the inverse. In this paper, we present a novel iterative algorithm (IBMI), which is designed to approximate the inverse of a large, dense, symmetric positive definite matrix. The matrix is first partitioned into blocks, and an iterative process using block matrix inversion is repeated until the matrix approximation reaches a satisfactory level of accuracy. We demonstrate that the two-block, non-overlapping approach converges for any positive definite matrix, while numerical results provide strong evidence that the multi-block, overlapping approach also converges for such matrices.