We show how to obliviously embed into the reproducing kernel Hilbert spaceassociated with Gaussian kernels, so that distance in this space (the kernel distance) only has (1+\varepsilon)-relative error . This only holds in comparing any point sets at a kernel distance at least \alpha ; this parameter only shows up as a poly-logarithmic factor of the dimension of an intermediate embedding, but not in the final embedding. The main insight is to effectively modify the well-traveled random Fourier features to be slightly biased and have higher variance, but so they can be defined as a convolution over the function space. This result provides the first guaranteed algorithmic results for LSH of kernel distance on point sets and low-dimensional shapes and distributions, and for relative error bounds on the kernel two-sample test .
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 .
R
show how to obliviously embed into the reproducing kernel Hilbert spaceassociated with Gaussian kernels, so that distance in this space (the kernel distance) only has
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
R
. This only holds in comparing any point sets at a kernel distance at least \alpha ; this parameter only shows up as a poly-logarithmic factor of the dimension of an intermediate embedding, but not in the final embedding. The main insight is to effectively modify the well-traveled random Fourier features to be slightly biased and have higher variance, but so they can be defined as a convolution over the function space. This result provides the first guaranteed algorithmic results for LSH of kernel distance on point sets and low-dimensional shapes and distributions, and for relative error bounds on the kernel two-sample test
, 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