Relative Error RKHS Embeddings for Gaussian Kernels
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We show how to obliviously embed into the reproducing kernel Hilbert space associated with Gaussian kernels, so that distance in this space (the kernel distance) only has (1+ε)-relative error. This only holds in comparing any point sets at a kernel distance at least α; 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.
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