Linear Time Kernel Matrix Approximation via Hyperspherical Harmonics

We propose a new technique for constructing low-rank approximations of matrices that arise in kernel methods for machine learning. Our approach pairs a novel automatically constructed analytic expansion of the underlying kernel function with a data-dependent compression step to further optimize the approximation. This procedure works in linear time and is applicable to any isotropic kernel. Moreover, our method accepts the desired error tolerance as input, in contrast to prevalent methods which accept the rank as input. Experimental results show our approach compares favorably to the commonly used Nystrom method with respect to both accuracy for a given rank and computational time for a given accuracy across a variety of kernels, dimensions, and datasets. Notably, in many of these problem settings our approach produces near-optimal low-rank approximations. We provide an efficient open-source implementation of our new technique to complement our theoretical developments and experimental results.

The Fast Kernel Transform

Kernel methods are a highly effective and widely used collection of modern machine learning algorithms. A fundamental limitation of virtually all such methods are computations involving the kernel matrix that naively scale quadratically (e.g., constructing the kernel matrix and matrix-vector multiplication) or cubically (solving linear systems) with the size of the data set $N.$ We propose the Fast Kernel Transform (FKT), a general algorithm to compute matrix-vector multiplications (MVMs) for datasets in moderate dimensions with quasilinear complexity. Typically, analytically grounded fast multiplication methods require specialized development for specific kernels. In contrast, our scheme is based on auto-differentiation and automated symbolic computations that leverage the analytical structure of the underlying kernel. This allows the FKT to be easily applied to a broad class of kernels, including Gaussian, Matern, and Rational Quadratic covariance functions and physically motivated Green's functions, including those of the Laplace and Helmholtz equations. Furthermore, the FKT maintains a high, quantifiable, and controllable level of accuracy -- properties that many acceleration methods lack. We illustrate the efficacy and versatility of the FKT by providing timing and accuracy benchmarks and by applying it to scale the stochastic neighborhood embedding (t-SNE) and Gaussian processes to large real-world data sets.