Papers › Fast Matrix Square Roots with Applications to Gaussian Processes and Bayesian Optimization

Fast Matrix Square Roots with Applications to Gaussian Processes and Bayesian Optimization

19 Jun 2020NeurIPS 2020 12arXiv:2006.11267archive 2025-07-28

Geoff Pleiss, Martin Jankowiak, David Eriksson, Anil Damle, Jacob R. Gardner

Matrix square roots and their inverses arise frequently in machine learning, e.g., when sampling from high-dimensional Gaussians 𝒩(0, 𝐊) or whitening a vector 𝐛 against covariance matrix 𝐊. While existing methods typically require O(N³) computation, we introduce a highly-efficient quadratic-time algorithm for computing 𝐊^(1/2) 𝐛, 𝐊^(-1/2) 𝐛, and their derivatives through matrix-vector multiplication (MVMs). Our method combines Krylov subspace methods with a rational approximation and typically achieves $4$ decimal places of accuracy with fewer than $100$ MVMs. Moreover, the backward pass requires little additional computation. We demonstrate our method's applicability on matrices as large as 50,000 ×50,000 - well beyond traditional methods - with little approximation error. Applying this increased scalability to variational Gaussian processes, Bayesian optimization, and Gibbs sampling results in more powerful models with higher accuracy.

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