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The first order derivative of a data density can be estimated efficiently bydenoising score matching, and has become an important component in manyapplications, such as image generation and audio synthesis. Higher orderderivatives provide additional local information about the data distributionand enable new applications. Although they can be estimated via automaticdifferentiation of a learned density model, this can amplify estimation errorsand is expensive in high dimensional settings. To overcome these limitations,we propose a method to directly estimate high order derivatives (scores) of adata density from samples. We first show that denoising score matching can beinterpreted as a particular case of Tweedie's formula. By leveraging Tweedie'sformula on higher order moments, we generalize denoising score matching toestimate higher order derivatives. We demonstrate empirically that modelstrained with the proposed method can approximate second order derivatives moreefficiently and accurately than via automatic differentiation. We show that ourmodels can be used to quantify uncertainty in denoising and to improve themixing speed of Langevin dynamics via Ozaki discretization for samplingsynthetic data and natural images.

Estimating High Order Gradients of the Data Distribution by Denoising