Kernel Interpolation with Sparse Grids
Mohit Yadav Daniel Sheldon Cameron Musco
Structured kernel interpolation (SKI) accelerates Gaussian process (GP)
inference by interpolating the kernel covariance function using a dense grid of
inducing points, whose corresponding kernel matrix is highly structured and
thus amenable to fast linear algebra. Unfortunately, SKI scales poorly in the
dimension of the input points, since the dense grid size grows exponentially
with the dimension. To mitigate this issue, we propose the use of sparse grids
within the SKI framework. These grids enable accurate interpolation, but with a
number of points growing more slowly with dimension. We contribute a novel
nearly linear time matrix-vector multiplication algorithm for the sparse grid
kernel matrix. Next, we describe how sparse grids can be combined with an
efficient interpolation scheme based on simplices. With these changes, we
demonstrate that SKI can be scaled to higher dimensions while maintaining
accuracy.