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Matrix square roots and their inverses arise frequently in machine learning,e.g., when sampling from high-dimensional Gaussians $\mathcal{N}(\mathbf 0,\mathbf K)$ or whitening a vector $\mathbf b$ against covariance matrix$\mathbf K$. While existing methods typically require $O(N^3)$ computation, weintroduce a highly-efficient quadratic-time algorithm for computing $\mathbfK^{1/2} \mathbf b$, $\mathbf K^{-1/2} \mathbf b$, and their derivatives throughmatrix-vector multiplication (MVMs). Our method combines Krylov subspacemethods with a rational approximation and typically achieves $4$ decimal placesof accuracy with fewer than $100$ MVMs. Moreover, the backward pass requireslittle additional computation. We demonstrate our method's applicability onmatrices as large as $50,\!000 \times 50,\!000$ - well beyond traditionalmethods - with little approximation error. Applying this increased scalabilityto variational Gaussian processes, Bayesian optimization, and Gibbs samplingresults in more powerful models with higher accuracy.

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