The Normalized Cross Density Functional: A Framework to Quantify Statistical Dependence for Random Processes

This paper proposes a novel multivariate definition of statistical dependence between two continuous random processes (r.p.) using a functional methodology inspired by Alfr\'ed R\'enyi. The argument of the logarithm of mutual information between pairs of samples of a r.p., named here the normalized cross density (NCD), defines a symmetric and self-adjoint positive definite function. We show that maximizing the alternating covariance estimation (ACE) recursion, applied to each of the joint probability density of input sample pairs, obeys all the properties of Renyi's maximal correlation. We propose the NCD's eigenspectrum as a novel multivariate measure of the statistical dependence between the input and output r.p. The multivariate statistical dependence can also be estimated directly from r.p. realizations. The proposed functional maximum correlation algorithm (FMCA) is applied to a machine learning architecture built from two neural networks that learn concurrently by approximating each others' outputs. We prove that the FMCA optimal solution is an equilibrium point that estimates the eigenspectrum of the cross density kernel. Preliminary results with synthetic data and medium size image datasets corroborate the theory. Four different strategies of applying the cross density kernel are proposed and thoroughly discussed to show the versatility and stability of the methodology, which transcends supervised learning. More specifically, when the two random processes are high-dimensional real-world images and a white uniform noise process, the algorithm learns a factorial code i.e., the occurrence of a code guarantees that a certain input in the training image set was present, which is quite important for feature learning.