Konstantin Yakovlev

Konstantin Yakovlev, Nikita Puchkin

We present a theory for simultaneous approximation of the score function and its derivatives, enabling the handling of data distributions with low-dimensional structure and unbounded support. Our approximation error bounds match those in the literature while relying on assumptions that relax the usual bounded support requirement. Crucially, our bounds are free from the curse of dimensionality. Moreover, we establish approximation guarantees for derivatives of any prescribed order, extending beyond the commonly considered first-order setting.

Konstantin Yakovlev, Anna Markovich, Nikita Puchkin

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 $L^2$-norms of smooth functions and their derivatives.