Gradient of Probability Density Functions based Contrasts for Blind Source Separation (BSS)

The article derives some novel independence measures and contrast functions for Blind Source Separation (BSS) application. For the $k^{th}$ order differentiable multivariate functions with equal hyper-volumes (region bounded by hyper-surfaces) and with a constraint of bounded support for $k>1$, it proves that equality of any $k^{th}$ order derivatives implies equality of the functions. The difference between product of marginal Probability Density Functions (PDFs) and joint PDF of a random vector is defined as Function Difference (FD) of a random vector. Assuming the PDFs are $k^{th}$ order differentiable, the results on generalized functions are applied to the independence condition. This brings new sets of independence measures and BSS contrasts based on the $L^p$-Norm, $ p \geq 1$ of - FD, gradient of FD (GFD) and Hessian of FD (HFD). Instead of a conventional two stage indirect estimation method for joint PDF based BSS contrast estimation, a single stage direct estimation of the contrasts is desired. The article targets both the efficient estimation of the proposed contrasts and extension of the potential theory for an information field. The potential theory has a concept of reference potential and it is used to derive closed form expression for the relative analysis of potential field. Analogous to it, there are introduced concepts of Reference Information Potential (RIP) and Cross Reference Information Potential (CRIP) based on the potential due to kernel functions placed at selected sample points as basis in kernel methods. The quantities are used to derive closed form expressions for information field analysis using least squares. The expressions are used to estimate $L^2$-Norm of FD and $L^2$-Norm of GFD based contrasts.