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

This paper presents a novel approach to measuring statistical dependence between two random processes (r.p.) using a positive-definite function called the Normalized Cross Density (NCD). NCD is derived directly from the probability density functions of two r.p. and constructs a data-dependent Hilbert space, the Normalized Cross-Density Hilbert Space (NCD-HS). By Mercer's Theorem, the NCD norm can be decomposed into its eigenspectrum, which we name the Multivariate Statistical Dependence (MSD) measure, and their sum, the Total Dependence Measure (TSD). Hence, the NCD-HS eigenfunctions serve as a novel embedded feature space, suitable for quantifying r.p. statistical dependence. In order to apply NCD directly to r.p. realizations, we introduce an architecture with two multiple-output neural networks, a cost function, and an algorithm named the Functional Maximal Correlation Algorithm (FMCA). With FMCA, the two networks learn concurrently by approximating each other's outputs, extending the Alternating Conditional Expectation (ACE) for multivariate functions. We mathematically prove that FMCA learns the dominant eigenvalues and eigenfunctions of NCD directly from realizations. Preliminary results with synthetic data and medium-sized image datasets corroborate the theory. Different strategies for applying NCD are proposed and discussed, demonstrating the method's versatility and stability beyond supervised learning. Specifically, when the two r.p. are high-dimensional real-world images and a white uniform noise process, FMCA learns factorial codes, i.e., the occurrence of a code guarantees that a specific training set image was present, which is important for feature learning.