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Beyond singular value gaps in randomized subspace approximation

Christopher Wang, Alex Townsend

TL;DR

This work shows that the so-called Frobenius singular value ratio provides a sharper analysis of an RRF's subspace quality under Gaussian sketching, and derives 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.

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.

Beyond singular value gaps in randomized subspace approximation

TL;DR

This work shows that the so-called Frobenius singular value ratio provides a sharper analysis of an RRF's subspace quality under Gaussian sketching, and derives 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.

Abstract

The success of randomized range finders (RRFs) is typically analyzed via the singular value gaps of a target matrix . 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 and any integer , we derive an explicit, closed-form expression for the cumulative distribution function of the largest principal angle between the -dominant singular subspace of 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.
Paper Structure (19 sections, 12 theorems, 72 equations, 6 figures, 1 algorithm)

This paper contains 19 sections, 12 theorems, 72 equations, 6 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Contributions
  3. Outline
  4. Background
  5. Principal angles
  6. Matrix spaces and probability measures
  7. Zonal polynomials and hypergeometric functions of matrix argument
  8. Distribution function of the RRF approximation error
  9. Heuristics
  10. Numerical experiments
  11. Probabilistic error bounds for the RRF
  12. Initial observations
  13. Probability estimates in terms of the Frobenius singular value ratio
  14. Worst-case gap analysis
  15. Numerical experiments
  16. ...and 4 more sections

Key Result

Theorem 3.1

Run the RRF (see alg:RRF) on a matrix $A\in\mathbb R^{m\times n}$ with $\mathop{\mathrm{rank}}\nolimits(A)=r$ and $k,p\ge0$. Under partition eq:A-partition, let $\theta_1$ be the largest principal angle between the true $k$-dominant left singular subspace and the RRF approximation. Then:

Figures (6)

  • Figure 1: Full-rank: $\sigma_j(A)=j^{-2}$, $1\le j\le100$.
  • Figure 1: Slow-decaying singular values, $p=1$.
  • Figure 1: Slow-decaying singular values, $p=5$.
  • Figure 2: Rank-deficient: $\sigma_j(A)=j^{-1}$, $1\le j\le2k+p-3$, and $\sigma_j(A)=0$, $j>2k+p-3$.
  • Figure 4: Slow-decaying singular values, $p=5$.
  • ...and 1 more figures

Theorems & Definitions (25)

  • Theorem 3.1
  • Lemma 3.2
  • Proof 1
  • Lemma 3.3
  • Proof 2
  • Lemma 3.4
  • Proof 3
  • Proof 4: Proof of \ref{['thm:cdf']}
  • Lemma 4.1
  • Proof 5
  • ...and 15 more