Estimating High Order Gradients of the Data Distribution by Denoising

TL;DR

This paper introduces a direct method to estimate high order derivatives of data distributions from samples, improving accuracy and efficiency over automatic differentiation, with applications in uncertainty quantification and sampling.

Contribution

It generalizes denoising score matching to estimate higher order derivatives using Tweedie's formula, offering a more efficient alternative to automatic differentiation.

Findings

Models accurately estimate second order derivatives.

Method improves sampling efficiency in high-dimensional data.

Enhanced uncertainty quantification in denoising tasks.

Abstract

The first order derivative of a data density can be estimated efficiently by denoising score matching, and has become an important component in many applications, such as image generation and audio synthesis. Higher order derivatives provide additional local information about the data distribution and enable new applications. Although they can be estimated via automatic differentiation of a learned density model, this can amplify estimation errors and is expensive in high dimensional settings. To overcome these limitations, we propose a method to directly estimate high order derivatives (scores) of a data density from samples. We first show that denoising score matching can be interpreted as a particular case of Tweedie's formula. By leveraging Tweedie's formula on higher order moments, we generalize denoising score matching to estimate higher order derivatives. We demonstrate empirically that models trained with the proposed method can approximate second order derivatives more efficiently and accurately than via automatic differentiation. We show that our models can be used to quantify uncertainty in denoising and to improve the mixing speed of Langevin dynamics via Ozaki discretization for sampling synthetic data and natural images.