Log-density Hessian estimation without the curse of dimensionality via denoising score matching

We study the problem of estimating the score function and its Jacobian matrix using denoising score matching. Assuming that the data distribution exhibits a low-dimensional structure, we prove that denoising score matching is able to estimate log-density Hessian 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 $L^2$-norms of smooth functions and their derivatives.