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arXiv · 2503.12966

Optimal Denoising in Score-Based Generative Models: The Role of Data Regularity

Eliot Beyler·Francis Bach
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Abstract

Score-based generative models achieve state-of-the-art sampling performance by denoising a distribution perturbed by Gaussian noise. In this paper, we focus on a single deterministic denoising step, and compare the optimal denoiser for the quadratic loss, we name ''full-denoising'', to the alternative ''half-denoising'' introduced by Hyv{\"a}rinen (2025). We show that looking at the performance in terms of distance between distributions tells a more nuanced story, with different assumptions on the data leading to very different conclusions. We prove that half-denoising is better than full-denoising for regular enough densities, while full-denoising is better for singular densities such as mixtures of Dirac measures or densities supported on a low-dimensional subspace. In the latter case, we prove that full-denoising can alleviate the curse of dimensionality under a linear manifold hypothesis.

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Eliot Beyler, Francis Bach. 2025-03-17. Optimal Denoising in Score-Based Generative Models: The Role of Data Regularity. https://arxiv.org/abs/2503.12966

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