Sampling and Identity-Testing Without Approximate Tensorization of Entropy

We study the problems of approximate sampling from and distribution testing of \emph{mixture models}, where the modes satisfy a functional inequality called \textit{approximate tensorization of entropy} (ATE). While it is known that ATE makes these tasks more efficient in the unimodal setting, mixtures of few distributions satisfying ATE do not necessarily satisfy ATE overall, leading to a lack of theoretical guarantees for multimodal distributions, which are a key challenging case of modern generative models. We show this gap can be overcome by establishing the following pair of results for mixtures of ATE distributions: 1) We show fast mixing of Glauber dynamics from a \textit{data-based initialization}, with \textit{optimal} sample complexity, for mixtures of distributions satisfying modified log-Sobolev inequalities, building on similar results in \cite{KoehlerV24, HuangMRW24} for mixtures satisfying the weaker Poincaré inequality. 2) Answering an open question from \cite{blanca2023complexity}, we give efficient identity-testers for mixtures of ATE distributions in the coordinate-conditional sampling access model.