Generalized Bregman Divergence and Gradient of Mutual Information for Vector Poisson Channels

TL;DR

This paper extends the gradient of mutual information and Bregman divergence concepts from scalar to vector Poisson channels, unifying analysis across different channel models and enabling applications like compressive sensing.

Contribution

It introduces a generalized Bregman divergence and extends the gradient of mutual information to vector Poisson channels, unifying scalar, vector, Gaussian, and Poisson models.

Findings

Generalized Bregman divergence exhibits properties similar to classical divergence.

Gradient of mutual information is derived for vector Poisson channels.

Framework applicable to X-ray and document classification tasks.

Abstract

We investigate connections between information-theoretic and estimation-theoretic quantities in vector Poisson channel models. In particular, we generalize the gradient of mutual information with respect to key system parameters from the scalar to the vector Poisson channel model. We also propose, as another contribution, a generalization of the classical Bregman divergence that offers a means to encapsulate under a unifying framework the gradient of mutual information results for scalar and vector Poisson and Gaussian channel models. The so-called generalized Bregman divergence is also shown to exhibit various properties akin to the properties of the classical version. The vector Poisson channel model is drawing considerable attention in view of its application in various domains: as an example, the availability of the gradient of mutual information can be used in conjunction with gradient descent methods to effect compressive-sensing projection designs in emerging X-ray and document classification applications.