Anna Markovich

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.

Anna Markovich, Nikita Puchkin

We propose an algorithm for nonparametric online change point detection based on sequential score function estimation and the tracking the best expert approach. The core of the procedure is a version of the fixed share forecaster tailored to the case of infinite number of experts and quadratic loss functions. The algorithm shows promising results in numerical experiments on artificial and real-world data sets. Its performance is supported by rigorous high-probability bounds describing behaviour of the test statistic in the pre-change and post-change regimes.