Memory Efficient And Minimax Distribution Estimation Under Wasserstein Distance Using Bayesian Histograms
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We study Bayesian histograms for distribution estimation on [0,1]^d under the Wasserstein W_v, 1 ≤ v < ∞ distance in the i.i.d sampling regime. We newly show that when d < 2v, histograms possess a special memory efficiency property, whereby in reference to the sample size n, order n^d/2v bins are needed to obtain minimax rate optimality. This result holds for the posterior mean histogram and with respect to posterior contraction: under the class of Borel probability measures and some classes of smooth densities. The attained memory footprint overcomes existing minimax optimal procedures by a polynomial factor in n; for example an n^1 - d/2v factor reduction in the footprint when compared to the empirical measure, a minimax estimator in the Borel probability measure class. Additionally constructing both the posterior mean histogram and the posterior itself can be done super–linearly in n. Due to the popularity of the W_1,W_2 metrics and the coverage provided by the d < 2v case, our results are of most practical interest in the (d=1,v =1,2), (d=2,v=2), (d=3,v=2) settings and we provide simulations demonstrating the theory in several of these instances.
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