Generic MANOVA limit theorems for products of projections

TL;DR

This paper establishes limit theorems for the spectral distribution of products of projection matrices, proving convergence to Wachter's MANOVA law and analyzing eigenvalue behavior using free probability and moment methods.

Contribution

It generalizes previous results on spectral convergence of projection products, proves a conjecture on random tight frames, and introduces new moment-based techniques for eigenvalue analysis.

Findings

Spectral distribution of projection products converges to Wachter's MANOVA law.

Largest eigenvalue converges to the law's right edge under certain conditions.

New moment recursion aids in analyzing eigenvalue convergence.

Abstract

We study the convergence of the empirical spectral distribution of $\mathbf{A} \mathbf{B} \mathbf{A}$ for $N \times N$ orthogonal projection matrices $\mathbf{A}$ and $\mathbf{B}$, where $\frac{1}{N}\mathrm{Tr}(\mathbf{A})$ and $\frac{1}{N}\mathrm{Tr}(\mathbf{B})$ converge as $N \to \infty$, to Wachter's MANOVA law. Using free probability, we show mild sufficient conditions for convergence in moments and in probability, and use this to prove a conjecture of Haikin, Zamir, and Gavish (2017) on random subsets of unit-norm tight frames. This result generalizes previous ones of Farrell (2011) and Magsino, Mixon, and Parshall (2021). We also derive an explicit recursion for the difference between the empirical moments $\frac{1}{N}\mathrm{Tr}((\mathbf{A} \mathbf{B} \mathbf{A})^k)$ and the limiting MANOVA moments, and use this to prove a sufficient condition for convergence in probability of the largest eigenvalue of $\mathbf{A} \mathbf{B} \mathbf{A}$ to the right edge of the support of the limiting law in the special case where that law belongs to the Kesten-McKay family. As an application, we give a new proof of convergence in probability of the largest eigenvalue when $\mathbf{B}$ is unitarily invariant; equivalently, this determines the limiting operator norm of a rectangular submatrix of size $\frac{1}{2}N \times \alpha N$ of a Haar-distributed $N \times N$ unitary matrix for any $\alpha \in (0, 1)$. Unlike previous proofs, we use only moment calculations and non-asymptotic bounds on the unitary Weingarten function, which we believe should pave the way to analyzing the largest eigenvalue for products of random projections having other distributions.