Reweighted information inequalities
This work addresses theoretical challenges in analyzing multimodal distributions, particularly for Langevin Monte Carlo, but appears incremental as it builds on existing log-Sobolev and transport-information inequalities.
The paper tackles the problem of establishing information inequalities for mixture distributions, showing that if a probability measure's components satisfy certain inequalities, then any measure close in relative Fisher information is also close in relative entropy or transport distance to a reweighted version of the mixture, providing a user-friendly interpretation for non-log-concave measures.
We establish a variant of the log-Sobolev and transport-information inequalities for mixture distributions. If a probability measure $Ï$ can be decomposed into components that individually satisfy such inequalities, then any measure $μ$ close to $Ï$ in relative Fisher information is close in relative entropy or transport distance to a reweighted version of $Ï$ with the same mixture components but possibly different weights. This provides a user-friendly interpretation of Fisher information bounds for non-log-concave measures and explains phenomena observed in the analysis of Langevin Monte Carlo for multimodal distributions.