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The article derives some novel independence measures and contrast functionsfor Blind Source Separation (BSS) application. For the $k^{th}$ orderdifferentiable multivariate functions with equal hyper-volumes (region boundedby hyper-surfaces) and with a constraint of bounded support for $k>1$, itproves that equality of any $k^{th}$ order derivatives implies equality of thefunctions. The difference between product of marginal Probability DensityFunctions (PDFs) and joint PDF of a random vector is defined as FunctionDifference (FD) of a random vector. Assuming the PDFs are $k^{th}$ orderdifferentiable, the results on generalized functions are applied to theindependence condition. This brings new sets of independence measures and BSScontrasts 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 indirectestimation method for joint PDF based BSS contrast estimation, a single stagedirect estimation of the contrasts is desired. The article targets both theefficient estimation of the proposed contrasts and extension of the potentialtheory for an information field. The potential theory has a concept ofreference potential and it is used to derive closed form expression for therelative analysis of potential field. Analogous to it, there are introducedconcepts of Reference Information Potential (RIP) and Cross ReferenceInformation Potential (CRIP) based on the potential due to kernel functionsplaced at selected sample points as basis in kernel methods. The quantities areused to derive closed form expressions for information field analysis usingleast squares. The expressions are used to estimate $L^2$-Norm of FD and$L^2$-Norm of GFD based contrasts.

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