Optimal Score Estimation Via Empirical Bayes Smoothing

Andre Wibisono,Yihong Wu,Kaylee Yingxi Yang

arXiv (Cornell University)（2024）

Yale University Department of Computer Science | Yale University Department of Statistics and Data Science

Cited 8|Views16

Abstract

We study the problem of estimating the score function of an unknownprobability distribution ρ^* from n independent and identicallydistributed observations in d dimensions. Assuming that ρ^* issubgaussian and has a Lipschitz-continuous score function s^*, we establishthe optimal rate of Θ̃(n^-2/d+4) for this estimationproblem under the loss function ŝ - s^*^2_L^2(ρ^*) that iscommonly used in the score matching literature, highlighting the curse ofdimensionality where sample complexity for accurate score estimation growsexponentially with the dimension d. Leveraging key insights in empiricalBayes theory as well as a new convergence rate of smoothed empiricaldistribution in Hellinger distance, we show that a regularized score estimatorbased on a Gaussian kernel attains this rate, shown optimal by a matchingminimax lower bound. We also discuss the implication of our theory on thesample complexity of score-based generative models.

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Robust Estimation

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