2024/02/12 by Andre Wibisono, Yihong Wu, Wibisono, Andre +3 · 6 citations
Computer Science · Mathematics · #Advanced Statistical Methods and Models #FOS: Computer and information sciences #FOS: Mathematics #Gaussian Processes and Bayesian Inference #Machine Learning (stat.ML) #Statistical Methods and Inference #Statistics Theory (math.ST)
paper · pdf · doi:10.48550/arxiv.2402.07747
openalex publication_date 2024/02/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study the problem of estimating the score function of an unknown probability distribution ρ^* from n independent and identically distributed observations in d dimensions. Assuming that ρ^* is subgaussian and has a Lipschitz-continuous score function s^*, we establish the optimal rate of Θ(n-(2)/(d+4)) for this estimation problem under the loss function ‖ s - s^*‖2L2(ρ^*) that is commonly used in the score matching literature, highlighting the curse of dimensionality where sample complexity for accurate score estimation grows exponentially with the dimension d. Leveraging key insights in empirical Bayes theory as well as a new convergence rate of smoothed empirical distribution in Hellinger distance, we show that a regularized score estimator based on a Gaussian kernel attains this rate, shown optimal by a matching minimax lower bound. We also discuss extensions to estimating β-Hölder continuous scores with β≤ 1, as well as the implication of our theory on the sample complexity of score-based generative models.