Nonparametric Score Estimators

TL;DR

This paper unifies and analyzes nonparametric score estimators derived from samples, providing theoretical insights, new estimator constructions, and efficient iterative methods with practical benefits.

Contribution

It offers a unifying framework for kernel-based score estimators, analyzes their convergence, and introduces new iterative regularization methods with computational advantages.

Findings

Unified analysis of existing score estimators

Construction of new estimators with desirable properties

Proposed iterative regularization methods with fast convergence

Abstract

Estimating the score, i.e., the gradient of log density function, from a set of samples generated by an unknown distribution is a fundamental task in inference and learning of probabilistic models that involve flexible yet intractable densities. Kernel estimators based on Stein's methods or score matching have shown promise, however their theoretical properties and relationships have not been fully-understood. We provide a unifying view of these estimators under the framework of regularized nonparametric regression. It allows us to analyse existing estimators and construct new ones with desirable properties by choosing different hypothesis spaces and regularizers. A unified convergence analysis is provided for such estimators. Finally, we propose score estimators based on iterative regularization that enjoy computational benefits from curl-free kernels and fast convergence.