  For applications as varied as Bayesian neural networks, determinantal point
processes, elliptical graphical models, and kernel learning for Gaussian
processes (GPs), one must compute a log determinant of an $n \times n$ positive
definite matrix, and its derivatives - leading to prohibitive
$\mathcal{O}(n^3)$ computations. We propose novel $\mathcal{O}(n)$ approaches
to estimating these quantities from only fast matrix vector multiplications
(MVMs). These stochastic approximations are based on Chebyshev, Lanczos, and
surrogate models, and converge quickly even for kernel matrices that have
challenging spectra. We leverage these approximations to develop a scalable
Gaussian process approach to kernel learning. We find that Lanczos is generally
superior to Chebyshev for kernel learning, and that a surrogate approach can be
highly efficient and accurate with popular kernels.
