Randomized Joint Diagonalization of Symmetric Matrices

TL;DR

This paper introduces a randomized method for joint diagonalization of nearly commuting symmetric matrices, providing robust guarantees and demonstrating superior performance over existing methods.

Contribution

The paper proposes a simple randomized algorithm for joint diagonalization that is easy to implement and offers provable robustness guarantees.

Findings

RJD successfully diagonalizes nearly commuting matrices with high probability.

The method achieves an error bound proportional to the near-commutativity parameter.

Numerical experiments show RJD outperforms existing approaches on synthetic and real data.

Abstract

Given a family of nearly commuting symmetric matrices, we consider the task of computing an orthogonal matrix that nearly diagonalizes every matrix in the family. In this paper, we propose and analyze randomized joint diagonalization (RJD) for performing this task. RJD applies a standard eigenvalue solver to random linear combinations of the matrices. Unlike existing optimization-based methods, RJD is simple to implement and leverages existing high-quality linear algebra software packages. Our main novel contribution is to prove robust recovery: Given a family that is $\epsilon$-near to a commuting family, RJD jointly diagonalizes this family, with high probability, up to an error of norm O($\epsilon$). We also discuss how the algorithm can be further improved by deflation techniques and demonstrate its state-of-the-art performance by numerical experiments with synthetic and real-world data.