Procrustes Problems on Random Matrices

2025-10-05

AI Summary

This paper studies the perturbed Procrustes point-set alignment problem, which seeks an orthogonal transformation minimizing the matrix norm discrepancy between two observed point sets. Motivated by robust alignment requirements under random matrix models, we systematically analyze and compare the performance of spectral, Frobenius, and robust norms within a nonsmooth Riemannian optimization framework. Theoretical analysis and empirical experiments demonstrate that, in canonical settings—including low-dimensional alignment and random network hypothesis testing—the Frobenius norm achieves statistical accuracy comparable to the computationally more expensive spectral or robust norms, while substantially reducing optimization complexity and computational cost. This finding establishes a principled, efficient norm selection criterion for Procrustes-type problems, offering both theoretical guarantees and practical scalability.

Technology Category

• Search and Optimization: Non-convex Optimization
• Intelligent Robots: Learning & Optimization for ROB
• Computer Vision: Adversarial Attacks & Robustness

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Security and Privacy: Large-scale security measurements

Abstract

Meaningful comparison between sets of observations often necessitates alignment or registration between them, and the resulting optimization problems range in complexity from those admitting simple closed-form solutions to those requiring advanced and novel techniques. We compare different Procrustes problems in which we align two sets of points after various perturbations by minimizing the norm of the difference between one matrix and an orthogonal transformation of the other. The minimization problem depends significantly on the choice of matrix norm; we highlight recent developments in nonsmooth Riemannian optimization and characterize which choices of norm work best for each perturbation. We show that in several applications, from low-dimensional alignments to hypothesis testing for random networks, when Procrustes alignment with the spectral or robust norm is the appropriate choice, it is often feasible to replace the computationally more expensive spectral and robust minimizers with their closed-form Frobenius-norm counterpart. Our work reinforces the synergy between optimization, geometry, and statistics.

Problem

• Comparing aligned observation sets via Procrustes optimization problems
• Analyzing matrix norm choices for orthogonal transformation alignment
• Replacing expensive spectral minimizers with Frobenius-norm counterparts

Innovation

• Uses nonsmooth Riemannian optimization for Procrustes problems
• Compares matrix norms to optimize alignment under perturbations
• Replaces spectral norms with closed-form Frobenius solutions