A universal expectation bound on empirical projections of deformed random matrices

TL;DR

This paper establishes a universal upper bound on the expected difference between empirical and true projections of deformed random matrices, extending previous Gaussian matrix results with a new deterministic approach based on singular value analysis.

Contribution

It provides a novel universal expectation bound for empirical projections of deformed random matrices, generalizing prior Gaussian matrix theorems with a deterministic analysis method.

Findings

Universal upper bound on expectation of projection difference.

Extension of Gaussian matrix results to broader random matrices.

Deterministic approach based on largest singular value analysis.

Abstract

Let $C$ be a real-valued $M\times M$ matrix with singular values $\lambda_1\ge...\ge\lambda_M$ and $E$ a random matrix of centered i.i.d. entries with finite fourth moment. In this paper we give a universal upper bound on the expectation of $||\hat\pi_rX||_{S_2}^2-||\pi_rX||^2_{S_2}$, where $X:=C+E$ and $\hat\pi_r$ (resp. $\pi_r$) is a rank-$r$ projection maximizing the Hilbert-Schmidt norm $||\tilde\pi_rX||_{S_2}$ (resp. $||\tilde\pi_rC||_{S_2}$) over the set $\S_{M,r}$ of all orthogonal rank-$r$ projections. This result is a generalization of a theorem for Gaussian matrices due to Rohde (2012). Our approach differs substantially from the techniques of the mentioned article. We analyze $||\hat\pi_rX||_{S_2}^2-||\pi_rX||^2_{S_2}$ from a rather deterministic point of view by an upper bound on $||\hat\pi_rX||_{S_2}^2-||\pi_rX||^2_{S_2}$, whose randomness is totally determined by the largest singular value of $E$.