Singular Subspace Perturbation Bounds via Rectangular Random Matrix Diffusions

TL;DR

This paper derives new bounds on the Frobenius distance between top-$k$ singular subspaces of a matrix and its Gaussian-perturbed version, using stochastic calculus and diffusion processes, with applications in statistics and differential privacy.

Contribution

It introduces a novel approach using Dyson-Bessel diffusion processes to analyze singular subspace perturbations under Gaussian noise.

Findings

Expected Frobenius distance scales as O(rac{\u221a{d}}{ ext{spectral gap}} imes  ext{sqrt}(T))

Bounds improve previous results by a factor involving rac{ ext{sqrt}(m)}{ ext{sqrt}(d)}  ext{sqrt}(k)

Method applies stochastic calculus to track subspace evolution under noise.

Abstract

Given a matrix $A \in \mathbb{R}^{m\times d}$ with singular values $\sigma_1\geq \cdots \geq \sigma_d$, and a random matrix $G \in \mathbb{R}^{m\times d}$ with iid $N(0,T)$ entries for some $T>0$, we derive new bounds on the Frobenius distance between subspaces spanned by the top-$k$ (right) singular vectors of $A$ and $A+G$. This problem arises in numerous applications in statistics where a data matrix may be corrupted by Gaussian noise, and in the analysis of the Gaussian mechanism in differential privacy, where Gaussian noise is added to data to preserve private information. We show that, for matrices $A$ where the gaps in the top-$k$ singular values are roughly $\Omega(\sigma_k-\sigma_{k+1})$ the expected Frobenius distance between the subspaces is $\tilde{O}(\frac{\sqrt{d}}{\sigma_k-\sigma_{k+1}} \times \sqrt{T})$, improving on previous bounds by a factor of $\frac{\sqrt{m}}{\sqrt{d}} \sqrt{k}$. To obtain our bounds we view the perturbation to the singular vectors as a diffusion process -- the Dyson-Bessel process -- and use tools from stochastic calculus to track the evolution of the subspace spanned by the top-$k$ singular vectors.