Memory Efficient And Minimax Distribution Estimation Under Wasserstein Distance Using Bayesian Histograms

• URL: http://arxiv.org/abs/2307.10099v1
• Date: Wed, 19 Jul 2023 16:13:20 GMT
• Title: Memory Efficient And Minimax Distribution Estimation Under Wasserstein Distance Using Bayesian Histograms
• Authors: Peter Matthew Jacobs, Lekha Patel, Anirban Bhattacharya, Debdeep Pati
• Abstract summary: We show that when $d 2v$, histograms possess a special textitmemory efficiency property, in reference to the sample size $n, order $nd/2v$ bins are needed to obtain minimax rate optimality. The attained memory footprint overcomes existing minimax optimal procedures by a factor in $n$; for example an $n1 - d/2v$ factor reduction in the footprint when compared to the empirical measure, a minimax estimator in the Borel probability measure class.
• Score: 6.21295508577576
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We study Bayesian histograms for distribution estimation on $[0,1]^d$ under the Wasserstein $W_v, 1 \leq v < \infty$ distance in the i.i.d sampling regime. We newly show that when $d < 2v$, histograms possess a special \textit{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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