Joint Approximate Partial Diagonalization of Large Matrices

TL;DR

This paper introduces a new approach for large matrices that finds a common near eigenspace to approximate joint diagonalization, enabling efficient analysis of high-dimensional data.

Contribution

It proposes a novel method for partial joint diagonalization by first identifying a near eigenspace before diagonalizing within it, addressing large matrix challenges.

Findings

Effective approximation of joint diagonalization in large matrices

Potential applications demonstrated through illustrative examples

New algorithms for partial joint diagonalization introduced

Abstract

Given a set of $p$ symmetric (real) matrices, the Orthogonal Joint Diagonalization (OJD) problem consists of finding an orthonormal basis in which the representation of each of these $p$ matrices is as close as possible to a diagonal matrix. We argue that when the matrices are of large dimension, then the natural generalization of this problem is to seek an orthonormal basis of a certain subspace that is a near eigenspace for all the matrices in the set. We refer to this as the problem of ``partial joint diagonalization of matrices.'' The approach proposed first finds this approximate common near eigenspace and then proceeds to a joint diagonalization of the restrictions of the input matrices in this subspace. A few solution methods for this problem are proposed and illustrations of its potential applications are provided.