Beyond singular value gaps in randomized subspace approximation

TL;DR

This paper introduces a new analysis method for randomized range finders using Frobenius singular value ratios, providing sharper probabilistic guarantees for subspace approximation quality beyond traditional singular value gap approaches.

Contribution

It derives a closed-form distribution for the principal angle between true and approximate subspaces, improving theoretical guarantees for Gaussian sketching methods.

Findings

Frobenius singular value ratio offers a sharper analysis than traditional singular value gaps.

Explicit distribution of the principal angle is derived using hypergeometric functions.

Probabilistic guarantees for randomized range finders are strengthened.

Abstract

The success of randomized range finders (RRFs) is typically analyzed via the singular value gaps of a target matrix $A$. In this work, we show that the so-called Frobenius singular value ratio provides a sharper analysis of an RRF's subspace quality under Gaussian sketching. For any matrix $A$ and any integer $k\ge0$, we derive an explicit, closed-form expression for the cumulative distribution function of the largest principal angle between the $k$-dominant singular subspace of $A$ and the approximate RRF subspace, expressing it in terms of a hypergeometric function. We obtain definitive probabilistic guarantees for RRFs that are strictly stronger than those obtained previously.