The GaussianSketch for Almost Relative Error Kernel Distance

We introduce a two versions of a new sketch for approximately embedding the Gaussian kernel into Euclidean inner product space. These work by truncating infinite expansions of the Gaussian inner product, and carefully invoking the TensorSketch. After providing concentration and approximation properties of these sketches, we demonstrate them to approximate the kernel distance between points sets. These sketches yield almost $(1+\varepsilon)$-relative error, but with a small additive $\alpha$ term. In the first variants the dependence on $1/\alpha$ is logarithmic, but has a separate exponential dependence on the original dimension $d$. In the second variant, the dependence on $1/\alpha$ is still sub-polynomial, but the dependence on $d$ is linear.