Generalized Orthogonal Procrustes Problem under Arbitrary Adversaries

TL;DR

This paper analyzes the generalized orthogonal Procrustes problem, proposing semidefinite relaxation and iterative methods with algebraic guarantees for exact recovery under adversarial noise, applicable across diverse settings.

Contribution

It introduces a purely algebraic analysis of SDR and GPM for GOPP, providing guarantees without statistical assumptions, and explores the optimization landscape for low-rank factorization.

Findings

SDR recovers the least squares estimator exactly

GPM converges linearly to the global minimizer with proper initialization

The optimization landscape is free of spurious local minima under certain conditions

Abstract

The generalized orthogonal Procrustes problem (GOPP) plays a fundamental role in several scientific disciplines including statistics, imaging science and computer vision. Despite its tremendous practical importance, it is generally an NP-hard problem to find the least squares estimator. We study the semidefinite relaxation (SDR) and an iterative method named generalized power method (GPM) to find the least squares estimator, and investigate the performance under a signal-plus-noise model. We show that the SDR recovers the least squares estimator exactly and moreover the generalized power method with a proper initialization converges linearly to the global minimizer to the SDR, provided that the signal-to-noise ratio is large. The main technique follows from showing the nonlinear mapping involved in the GPM is essentially a local contraction mapping and then applying the well-known Banach fixed-point theorem finishes the proof. In addition, we analyze the low-rank factorization algorithm and show the corresponding optimization landscape is free of spurious local minimizers under nearly identical conditions that enables the success of SDR approach. The highlight of our work is that the theoretical guarantees are purely algebraic and do not assume any statistical priors of the additive adversaries, and thus it applies to various interesting settings.