I will present very preliminary work on a new approach to general integration which is an alternative to Markov chain sampling, largely considered to be the only practical option in the high-dimensional setting. The idea of the proposed method is to directly reconstruct the function through repeated nonparametric estimation and inference-guided sampling, with the aid of multiscale geometric data structures. The past attempts along these lines have not yielded practical alternatives to MCMC. I will argue both theoretically and experimentally that two key insights are needed to overcome the barrier to practicality, one statistical and one computational.
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