Local moment matching: A unified methodology for symmetric functional estimation and distribution estimation under Wasserstein distance
This work provides a unified methodology for distribution and functional estimation, addressing a fundamental challenge in statistics and machine learning with broad applications in data analysis.
The paper tackles the problem of estimating symmetric functionals and distributions under Wasserstein distance by introducing Local Moment Matching (LMM), which achieves minimax rates for distribution estimation and optimal sample complexity for entropy, power sum, and support size functionals.
We present \emph{Local Moment Matching (LMM)}, a unified methodology for symmetric functional estimation and distribution estimation under Wasserstein distance. We construct an efficiently computable estimator that achieves the minimax rates in estimating the distribution up to permutation, and show that the plug-in approach of our unlabeled distribution estimator is "universal" in estimating symmetric functionals of discrete distributions. Instead of doing best polynomial approximation explicitly as in existing literature of functional estimation, the plug-in approach conducts polynomial approximation implicitly and attains the optimal sample complexity for the entropy, power sum and support size functionals.