Abstract:We consider the problem of score function estimation via empirical risk minimization. We first start with the question of inferring the score function of a probability measure $\mu$ with density on the flat torus from a sample of distribution $\mu$. We show that constraining the hypothesis space to a Sobolev ball is sufficient to prevent overfitting and obtaining minimax estimation rates. We then consider the problem of score function estimation in the context of score-based generative modeling. Again, under a conjecture tying the score estimation rates to the quality of the output of a score-based generative model, we obtain minimax rates for such an approach using score function estimators obtained by constraining the hypothesis class to a Sobolev ball.
From: Thomas Bonis [view email]
[v1]
Wed, 17 Jun 2026 13:55:20 UTC (37 KB)
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