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{ä}rinen (2024). We show that looking at the performances in term of distance between distribution tells a more nuanced story, with different assumptions on the data leading to very differentthis http URLprove 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.
View on arXiv@article{beyler2025_2503.12966,
title={ Optimal Denoising in Score-Based Generative Models: The Role of Data Regularity },
author={ Eliot Beyler and Francis Bach },
journal={arXiv preprint arXiv:2503.12966},
year={ 2025 }
}