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 (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.

翻译：基于分数的生成模型通过去噪受高斯噪声扰动的分布，实现了最先进的采样性能。本文聚焦于单个确定性去噪步骤，比较了二次损失下的最优去噪器（我们称之为“全去噪”）与Hyvärinen（2025）提出的替代方案“半去噪”。我们证明，从分布间距离的角度评估性能会呈现更细致的情况，对数据的不同假设会导致截然不同的结论。我们证明了对于足够正则的密度，半去噪优于全去噪；而对于奇异密度（如狄拉克测度混合或支撑在低维子空间上的密度），全去噪则更优。在后一种情况下，我们证明了在线性流形假设下，全去噪能够缓解维数灾难问题。