Matrix compression along isogenic blocks

TL;DR

This paper introduces a novel matrix compression method based on isogenic block decomposition, preserving spectral properties and useful for large correlation matrices, with potential applications in various computational matrix analyses.

Contribution

It presents a new matrix compression algorithm using isogenic blocks, highlighting its functorial and spectral-permanence properties, and explores its applications in computational matrix analysis.

Findings

Preserves spectral properties during compression and inflation.

Applicable to large correlation matrices and hierarchical matrix structures.

Offers potential for improved matrix analysis techniques.

Abstract

A matrix-compression algorithm is derived from a novel isogenic block decomposition for square matrices. The resulting compression and inflation operations possess strong functorial and spectral-permanence properties. The basic observation that Hadamard entrywise functional calculus preserves isogenic blocks has already proved to be of paramount importance for thresholding large correlation matrices. The proposed isogenic stratification of the set of complex matrices bears similarities to the Schubert cell stratification of a homogeneous algebraic manifold. An array of potential applications to current investigations in computational matrix analysis is briefly mentioned, touching concepts such as symmetric statistical models, hierarchical matrices and coherent matrix organization induced by partition trees.