Orbital minimization method with $\ell^1$ regularization

TL;DR

This paper introduces an $ ext{l}^1$ regularized orbital minimization method to promote sparsity in eigenspace representations, analyzing its properties, proposing algorithms, and validating through numerical tests.

Contribution

It develops a modified OMM functional with $ ext{l}^1$ penalty, analyzes its minima, and proposes algorithms combining soft thresholding with gradient descent.

Findings

Every local minimum of the original OMM functional is also a global minimum.

The modified functional with $ ext{l}^1$ regularization effectively promotes sparsity.

Numerical tests confirm the effectiveness of the proposed algorithms.

Abstract

We consider a modification of the OMM energy functional which contains an $\ell^1$ penalty term in order to find a sparse representation of the low-lying eigenspace of self-adjoint operators. We analyze the local minima of the modified functional as well as the convergence of the modified functional to the original functional. Algorithms combining soft thresholding with gradient descent are proposed for minimizing this new functional. Numerical tests validate our approach. As an added bonus, we also prove the unanticipated and remarkable property that every local minimum the OMM functional without the $\ell^1$ term is also a global minimum.