Procrustes Problems on Random Matrices

TL;DR

This work analyzes Procrustes alignment problems under three matrix norms—Frobenius, spectral, and robust $\ell_{2,1}$—in the context of random matrices and network data. It shows that the Frobenius-norm solution $W_F^*$ often provides a computationally cheap proxy for the spectral and robust minimizers, with surprisingly little loss in inference power in many network-testing scenarios. Using Riemannian optimization on the orthogonal group and nonsmooth solvers, the authors compare test statistics $T_F$, $T_S$, and $T_R$ (and their Frobenius-minimizer variants) via bootstrapped critical values across diffuse, rank-one, and salt-and-pepper perturbations, revealing when each norm yields the strongest power. The findings support a practical stance: select the norm by the perturbation structure, and consider Frobenius-based substitutions to gain substantial computational savings without sacrificing accuracy in many hypothesis-testing tasks for networks and latent-position models.

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.

Figures (9)

• Figure 1: The two-dimensional alignment task comparing Frobenius, Spectral, and Robust Procrustes when outliers are present
• Figure 2: Orthogonal and arbitrary outliers.
• Figure 3: Two global dislocations affecting all points.
• Figure 4: The error in angle for the experiment $\tilde{D}$ visualized on the corresponding cost functions $f_n$ for the case of the Frobenius ($n=\mathrm{F}$) and Spectral ($n=\mathrm{S}$) as well as the cost of the Frobenius minimizer plugged into the spectral cost.
• Figure 5: Power curves for diffuse noise in dimension 3