  We introduce 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 kernel, and carefully invoking the
RecursiveTensorSketch [Ahle et al. SODA 2020]. After providing concentration
and approximation properties of these sketches, we use 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 poly-logarithmic, but has
higher degree of polynomial dependence on the original dimension $d$. In the
second variant, the dependence on $1/\alpha$ is still poly-logarithmic, but the
dependence on $d$ is linear.
