Accelerating Certifiable Estimation with Preconditioned Eigensolvers

TL;DR

This paper introduces a preconditioned eigensolver approach that significantly accelerates the verification step in certifiable estimation, enabling faster solutions for large-scale semidefinite relaxations in machine perception tasks.

Contribution

It develops a specialized preconditioned eigensolver using LOBPCG and an algebraic preconditioner to speed up eigenpair computations in certifiable estimation methods.

Findings

Achieves up to 280x speedup in verification time

Reduces overall Burer-Monteiro method runtime by up to 16x

Effective on simulated and real-world SLAM benchmarks

Abstract

Convex (specifically semidefinite) relaxation provides a powerful approach to constructing robust machine perception systems, enabling the recovery of certifiably globally optimal solutions of challenging estimation problems in many practical settings. However, solving the large-scale semidefinite relaxations underpinning this approach remains a formidable computational challenge. A dominant cost in many state-of-the-art (Burer-Monteiro factorization-based) certifiable estimation methods is solution verification (testing the global optimality of a given candidate solution), which entails computing a minimum eigenpair of a certain symmetric certificate matrix. In this letter, we show how to significantly accelerate this verification step, and thereby the overall speed of certifiable estimation methods. First, we show that the certificate matrices arising in the Burer-Monteiro approach generically possess spectra that make the verification problem expensive to solve using standard iterative eigenvalue methods. We then show how to address this challenge using preconditioned eigensolvers; specifically, we design a specialized solution verification algorithm based upon the locally optimal block preconditioned conjugate gradient (LOBPCG) method together with a simple yet highly effective algebraic preconditioner. Experimental evaluation on a variety of simulated and real-world examples shows that our proposed verification scheme is very effective in practice, accelerating solution verification by up to 280x, and the overall Burer-Monteiro method by up to 16x, versus the standard Lanczos method when applied to relaxations derived from large-scale SLAM benchmarks.