[1504.02712] Gradient of Probability Density Functions based Contrasts for Blind Source Separation (BSS)
The required performance analysis is restricted hereand targeted in the future versions of this article.

Abstract: 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.