Fast Matrix Square Roots with Applications to Gaussian Processes and Bayesian Optimization
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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(N^3) computation, we introduce a highly-efficient quadratic-time algorithm for computing K^1/2b, K^-1/2b, 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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