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

Matrix square roots and their inverses arise frequently in machine learning, e.g., when sampling from high-dimensional Gaussians N(mathbf 0, mathbf K) or whitening a vector mathbf b against covariance matrix mathbf K. While existing methods typically require O(N3) computation, we introduce a highly-efficient quadratic-time algorithm for computing mathbf K1/2 mathbf b, mathbf K-1/2 mathbf 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.