# Unifying the Brascamp-Lieb Inequality and the Entropy Power Inequality

## TL;DR

This paper introduces a new family of entropy functionals that unify and generalize the entropy power inequality and the Brascamp-Lieb inequality, revealing Gaussian extremality and intermediate inequalities.

## Contribution

It defines subadditive entropy functionals, proves Gaussian extremality, and derives a generalized inequality that encompasses both EPI and BLI.

## Key findings

- Gaussians are extremal for the new entropy functionals.
- A new inequality generalizes both EPI and BLI.
- Intermediate inequalities are obtained based on component independence.

## Abstract

The entropy power inequality (EPI) and the Brascamp-Lieb inequality (BLI) are fundamental inequalities concerning the differential entropies of linear transformations of random vectors. The EPI provides lower bounds for the differential entropy of linear transformations of random vectors with independent components. The BLI, on the other hand, provides upper bounds on the differential entropy of a random vector in terms of the differential entropies of some of its linear transformations. In this paper, we define a family of entropy functionals, which we show are subadditive. We then establish that Gaussians are extremal for these functionals by mimicking the idea in Geng and Nair (2014). As a consequence, we obtain a new entropy inequality that generalizes both the BLI and EPI. By considering a variety of independence relations among the components of the random vectors appearing in these functionals, we also obtain families of inequalities that lie between the EPI and the BLI.