Sharp detection of low-dimensional structure in probability measures via dimensional logarithmic Sobolev inequalities
This work addresses the challenge of dimension reduction in sampling for generative modeling, offering incremental improvements over prior methods.
The paper tackles the problem of identifying low-dimensional structure in high-dimensional probability measures for efficient sampling by introducing a method that approximates a target measure as a perturbation of a reference measure along significant directions. It shows that minimizing the dimensional logarithmic Sobolev inequality is equivalent to minimizing the Kullback-Leibler divergence for Gaussian measures and provides improved majorants for non-Gaussian cases.
Identifying low-dimensional structure in high-dimensional probability measures is an essential pre-processing step for efficient sampling. We introduce a method for identifying and approximating a target measure $π$ as a perturbation of a given reference measure $μ$ along a few significant directions of $\mathbb{R}^{d}$. The reference measure can be a Gaussian or a nonlinear transformation of a Gaussian, as commonly arising in generative modeling. Our method extends prior work on minimizing majorizations of the Kullback--Leibler divergence to identify optimal approximations within this class of measures. Our main contribution unveils a connection between the \emph{dimensional} logarithmic Sobolev inequality (LSI) and approximations with this ansatz. Specifically, when the target and reference are both Gaussian, we show that minimizing the dimensional LSI is equivalent to minimizing the KL divergence restricted to this ansatz. For general non-Gaussian measures, the dimensional LSI produces majorants that uniformly improve on previous majorants for gradient-based dimension reduction. We further demonstrate the applicability of this analysis to the squared Hellinger distance, where analogous reasoning shows that the dimensional Poincaré inequality offers improved bounds.