  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.
