Unifying the Brascamp-Lieb Inequality and the Entropy Power Inequality

• Published in: IEEE transactions on information theory Vol. 68; no. 12; pp. 7665 - 7684
• Main Authors: Anantharam, Venkat, Jog, Varun, Nair, Chandra
• Format: Journal Article
• Language: English
• Published: New York IEEE 01.12.2022; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Brascamp-Lieb inequality; Cramer-Rao bounds; Data processing; Entropy; Entropy power inequality; Gaussian distribution; Inequalities; Inequality; Linear matrix inequalities; Linear transformations; Lower bounds; Random variables; subadditivity; Sufficient conditions; Upper bounds
• ISSN: 0018-9448; 1557-9654
• DOI: 10.1109/TIT.2022.3192913

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 adapting a proof technique from 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.