[Paper Review] Randomly pivoted Cholesky: Practical approximation of a kernel matrix with few entry evaluations

Yifan Chen, Ethan N. Epperly+2 more|arXiv (Cornell University)|Jul 13, 2022

Sparse and Compressive Sensing Techniques4 citations

TL;DR

This paper introduces Randomly Pivoted Cholesky (RPCholesky), a practical algorithm for low-rank approximation of large positive-semidefinite kernel matrices using only O(kN) entry evaluations and O(k²N) operations. It achieves nearly optimal approximation error with high probability, outperforming existing methods in speed, accuracy, and robustness while requiring no parameter tuning.

ABSTRACT

The randomly pivoted partial Cholesky algorithm (RPCholesky) computes a factorized rank-k approximation of an N x N positive-semidefinite (psd) matrix. RPCholesky requires only (k + 1) N entry evaluations and O(k^2 N) additional arithmetic operations, and it can be implemented with just a few lines of code. The method is particularly useful for approximating a kernel matrix. This paper offers a thorough new investigation of the empirical and theoretical behavior of this fundamental algorithm. For matrix approximation problems that arise in scientific machine learning, experiments show that RPCholesky matches or beats the performance of alternative algorithms. Moreover, RPCholesky provably returns low-rank approximations that are nearly optimal. The simplicity, effectiveness, and robustness of RPCholesky strongly support its use in scientific computing and machine learning applications.

Motivation & Objective

• To address the computational bottleneck of full kernel matrix computation and storage in kernel methods for large-scale scientific machine learning.
• To develop a low-rank approximation algorithm that reveals only O(kN) entries of the kernel matrix while maintaining high accuracy.
• To ensure the method is robust, simple to implement, and requires no user-adjusted parameters.
• To provide theoretical guarantees on approximation error that are competitive with the best possible low-rank approximations.
• To empirically validate RPCholesky’s superiority over existing algorithms in speed, accuracy, and reliability on real-world scientific machine learning problems.

Proposed method

• RPCholesky adaptively selects pivot columns using a probabilistic rule based on the diagonal of the residual matrix, ensuring high-quality column sampling.
• At each iteration, it computes the residual column by subtracting the contribution of previously selected columns, then normalizes the pivot column.
• The algorithm maintains a running estimate of the diagonal of the residual matrix and updates it by subtracting the squared entries of the new factor column.
• The method returns a factored low-rank approximation  = FF* using only O(kN) storage and O(k²N) arithmetic operations.
• It leverages randomized pivot selection to avoid poor choices common in greedy or fixed pivot rules.
• Theoretical analysis uses matrix Bernstein inequalities to bound the spectral and trace norms of the approximation error.

Experimental results

Research questions

• RQ1 Can a low-rank approximation algorithm for kernel matrices be designed that requires only O(kN) entry evaluations and O(k²N) operations while maintaining high accuracy?
• RQ2 How does RPCholesky’s performance compare to other state-of-the-art matrix approximation algorithms in terms of speed, accuracy, and robustness?
• RQ3 What theoretical guarantees can be established for the spectral and trace norm error of RPCholesky’s low-rank approximation?
• RQ4 Does the randomized pivot selection rule ensure consistent performance across diverse kernel matrices, even when deterministic rules fail?
• RQ5 Can RPCholesky achieve approximation error that is nearly optimal, comparable to the best possible rank-r approximation?

Key findings

• RPCholesky achieves a spectral norm error bound of ‖A − Â‖ ≤ λ, where λ is a parameter controlling the approximation quality, ensuring near-optimality.
• The trace norm error is bounded by (1 + ε/2) · tr(A − [A]_r), meaning the expected error is within a small factor of the best possible rank-r approximation.
• Empirical results show RPCholesky matches or exceeds the performance of alternative algorithms in speed, accuracy, and reliability on scientific machine learning problems.
• The method is provably robust: failure occurs with probability at most εη/2, and even in failure cases, the trace error remains bounded by tr(A).
• The algorithm requires only (k+1)N entry evaluations and O(k²N) operations, making it highly efficient for large-scale kernel matrices.
• Theoretical analysis confirms that RPCholesky produces approximations that are nearly optimal in both spectral and trace norms, with high probability.