Maximizing the Smallest Eigenvalue of a Symmetric Matrix: A Submodular Optimization Approach

TL;DR

This paper introduces a submodular optimization method for selecting submatrices of positive definite matrices to maximize their smallest eigenvalue, with applications in network consensus and distributed systems.

Contribution

It develops a novel submodular framework for eigenvalue maximization, providing polynomial-time algorithms with theoretical guarantees.

Findings

Efficient algorithms for submatrix selection with eigenvalue bounds

Theoretical bounds on submatrix size and eigenvalue improvement

Numerical validation demonstrating effectiveness

Abstract

This paper studies the problem of selecting a submatrix of a positive definite matrix in order to achieve a desired bound on the smallest eigenvalue of the submatrix. Maximizing this smallest eigenvalue has applications to selecting input nodes in order to guarantee consensus of networks with negative edges as well as maximizing the convergence rate of distributed systems. We develop a submodular optimization approach to maximizing the smallest eigenvalue by first proving that positivity of the eigenvalues of a submatrix can be characterized using the probability distribution of the quadratic form induced by the submatrix. We then exploit that connection to prove that positive-definiteness of a submatrix can be expressed as a constraint on a submodular function. We prove that our approach results in polynomial-time algorithms with provable bounds on the size of the submatrix. We also present generalizations to non-symmetric matrices, alternative sufficient conditions for the smallest eigenvalue to exceed a desired bound that are valid for Laplacian matrices, and a numerical evaluation.