  In this paper, we develop a framework for information theoretic learning
based on infinitely divisible matrices. We formulate an entropy-like functional
on positive definite matrices based on Renyi's axiomatic definition of entropy
and examine some key properties of this functional that lead to the concept of
infinite divisibility. The proposed formulation avoids the plug in estimation
of density and brings along the representation power of reproducing kernel
Hilbert spaces. As an application example, we derive a supervised metric
learning algorithm using a matrix based analogue to conditional entropy
achieving results comparable with the state of the art.
