Paper ID sheet UCL-INMA-2012.09


Spherical mesh adaptive direct search for separating quasi-uncorrelated sources by range-based independent component analysis

S. Easter Selvan, Pierre B. Borckmans, Amit Chattopadhyay, P.-A. Absil
It is seemingly paradoxical to the classical definition of the independent component analysis (ICA), that in reality the true sources are often not strictly uncorrelated. With this in mind, this paper concerns a framework to extract quasi-uncorrelated sources with finite supports by optimizing a range-based contrast function under unit-norm constraints (to handle the inherent scaling indeterminacy of ICA) but without orthogonality constraints. Albeit the appealing contrast properties of the range-based function, e.g., the absence of mixing local optima, the function is not differentiable everywhere. Unfortunately, there is a dearth of literature on derivative-free optimizers that effectively handle such a nonsmooth yet promising contrast function. This is the compelling reason for the design of a nonsmooth optimization algorithm on a manifold of matrices having unit-norm columns with the following objectives: (i) to ascertain convergence to a Clarke stationary point of the contrast function; (ii) to adhere to the necessary unit-norm constraints more naturally. The proposed nonsmooth optimization algorithm crucially relies on the design and analysis of an extension of the Mesh Adaptive Direct Search (MADS) method to handle locally Lipschitz objective functions defined on the sphere. The applicability of the algorithm in the ICA domain is demonstrated with simulations involving natural, face, aerial and texture images.
Key words
Clarke stationary point; matrix manifold; nonsmooth optimizer; range-based contrast; source separation
Neural Computation, Vol. 25, No. 9, Pages 2486-2522, September 2013
BibTeX entry
author = "S. Easter Selvan and Pierre B. Borckmans and A. Chattopadhyay and P.-A. Absil",
title = "Spherical Mesh Adaptive Direct Search for Separating Quasi-Uncorrelated Sources by Range-Based Independent Component Analysis",
fjournal = "Neural Computation",
journal = "Neural Comput.",
volume = 25,
number = 9,
pages = "2486--2522",
year = 2013,