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We will discuss several topics related to the importance of using low frequency eigenfunctions of the Gaussian kernel. First, we will discuss the relevance of selecting important eigenfunctions for two sample testing and statistical distances, namely kernel Maximum Mean Discrepancy. This creates a more powerful test than the classical MMD while still maintaining sensitivity to common departures. Second, we derive an algorithm for sub-sampling smooth functions (i.e. functions that are linear combinations of the low frequency eigenfunctions) that provably upper bounds the error in estimating the mean of the function, and use this to bound the error from subsampled MMD. Finally, we will discuss an efficient method for sampling new points from the space spanned by the low frequency eigenfunctions, and use a deep learning framework similar to Variational Autoencdoers to map these new points back to the original data space in a way that respects the original data geometry.