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Randomly Pivoted Partial Cholesky: Random How?

Stefan Steinerberger

TL;DR

The paper addresses efficient low-rank approximation of symmetric positive-definite matrices using randomly pivoted partial Cholesky. It analyzes pivot strategies under a Gibbs-like scheme with exponent $\beta$, proving that sampling with probability proportional to $A_{ii}^2$ yields contraction in the Frobenius norm: $\mathbb{E}\|B\|_F^2 \le \|A\|_F^2 - \frac{1}{\sum_i A_{ii}^2}\sum_i \|A_i\|_2^4 \le (1 - 1/n)\|A\|_F^2$, with the bound sharp for diagonal $A$ and strengthened by column-norm fluctuations. The work also connects to greedy pivoting, provides bounds that justify large diagonal choices, and proposes alternating pivoting between greedy and random strategies to hedge against misleading diagonal cues. Through preparatory computations and a key inequality $A_{ii}(A^3)_{ii} \ge \|A_i\|_2^4$, the authors derive rigorous proofs of the main theorem and corollaries, situating the results within the broader literature on randomized low-rank matrix approximations and kernel methods. The findings offer practical guidance for pivot selection in large-scale SPD matrices where entry evaluations are costly. $ $

Abstract

We consider the problem of finding good low rank approximations of symmetric, positive-definite $A \in \mathbb{R}^{n \times n}$. Chen-Epperly-Tropp-Webber showed, among many other things, that the randomly pivoted partial Cholesky algorithm that chooses the $i-$th row with probability proportional to the diagonal entry $A_{ii}$ leads to a universal contraction of the trace norm (the Schatten 1-norm) in expectation for each step. We show that if one chooses the $i-$th row with likelihood proportional to $A_{ii}^2$ one obtains the same result in the Frobenius norm (the Schatten 2-norm). Implications for the greedy pivoting rule and pivot selection strategies are discussed.

Randomly Pivoted Partial Cholesky: Random How?

TL;DR

The paper addresses efficient low-rank approximation of symmetric positive-definite matrices using randomly pivoted partial Cholesky. It analyzes pivot strategies under a Gibbs-like scheme with exponent , proving that sampling with probability proportional to yields contraction in the Frobenius norm: , with the bound sharp for diagonal and strengthened by column-norm fluctuations. The work also connects to greedy pivoting, provides bounds that justify large diagonal choices, and proposes alternating pivoting between greedy and random strategies to hedge against misleading diagonal cues. Through preparatory computations and a key inequality , the authors derive rigorous proofs of the main theorem and corollaries, situating the results within the broader literature on randomized low-rank matrix approximations and kernel methods. The findings offer practical guidance for pivot selection in large-scale SPD matrices where entry evaluations are costly.

Abstract

We consider the problem of finding good low rank approximations of symmetric, positive-definite . Chen-Epperly-Tropp-Webber showed, among many other things, that the randomly pivoted partial Cholesky algorithm that chooses the th row with probability proportional to the diagonal entry leads to a universal contraction of the trace norm (the Schatten 1-norm) in expectation for each step. We show that if one chooses the th row with likelihood proportional to one obtains the same result in the Frobenius norm (the Schatten 2-norm). Implications for the greedy pivoting rule and pivot selection strategies are discussed.
Paper Structure (15 sections, 5 theorems, 37 equations, 6 figures, 1 table)

This paper contains 15 sections, 5 theorems, 37 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Randomly pivoted Cholesky.
  3. The case $\beta = 1$
  4. Results
  5. Frobenius norm.
  6. Greedy Pivoting
  7. Some examples.
  8. Alternating Pivot Selection.
  9. Related results
  10. Proofs
  11. Some preparatory computations.
  12. An Inequality.
  13. Proof of the Theorem.
  14. Proof of Corollary 1.
  15. Proof of Corollary 2.

Key Result

Theorem 1

Suppose $A \in \mathbb{R}^{n \times n}$ is spd and $B$ arises from $A$ by selecting the $i-$th row/column as pivot with likelihood proportional to $A_{ii}^2$. Then

Figures (6)

  • Figure 1: Size of $M^{(50)}$ relative to $M^{(0)} = A$ when $\beta=1$.
  • Figure 2: Size of $M^{(50)}$ relative to $M^{(0)} = A$ when $\beta=\infty$.
  • Figure 3: Points on a spiral.
  • Figure 4: The matrix $A$.
  • Figure 5: Size of $M^{(20)}$ relative to $M^{(0)} = A$ when $f(i) = 1/i$.
  • ...and 1 more figures

Theorems & Definitions (10)

  • Theorem
  • Corollary 1
  • Corollary 2
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • proof
  • proof
  • proof