# Implicit score matching meets denoising score matching: improved rates of convergence and log-density Hessian estimation

arXiv: https://arxiv.org/abs/2512.24378 · HTML: https://mlsanity.com/paper/2512.24378/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 30 Dec 2025.

## Built on by (most cited first, 4 of 4)

- [Inference-Time Alignment for Diffusion Models via Variationally Stable Doob's Matching](https://mlsanity.com/paper/2601.06514/) (2026)
- [Diffusion Models with Heavy-Tailed Targets: Score Estimation and Sampling Guarantees](https://mlsanity.com/paper/2601.06715/) (2026)
- [Finite Sample Bounds for Learning with Score Matching](https://mlsanity.com/paper/2605.14168/) (2026)
- [Robustness of Diffusion Models under Distribution Shift](https://mlsanity.com/paper/2609.27546/) (2026)

## Abstract (arXiv)

We study the problem of estimating the score function using both implicit score matching and denoising score matching. Assuming that the data distribution exhibiting a low-dimensional structure, we prove that implicit score matching is able not only to adapt to the intrinsic dimension, but also to achieve the same rates of convergence as denoising score matching in terms of the sample size. Furthermore, we demonstrate that both methods allow us to estimate log-density Hessians without the curse of dimensionality by simple differentiation. This justifies convergence of ODE-based samplers for generative diffusion models. Our approach is based on Gagliardo-Nirenberg-type inequalities relating weighted L2-norms of smooth functions and their derivatives.