Spectral Independent Component Analysis

Independent component analysis (ICA) of a
mixed signal into a linear combination of its independent
components, is one of the main problems in statistics, with wide
range of applications. The un-mixing is usually performed by finding
a rotation that optimizes a functional closely related to the
differential entropy. In this talk we solve the linear ICA problem
by analyzing the spectrum and eigenspaces of the graph laplacian of
the data. The spectral ICA algorithm is based on two observations.
First, independence of random variables is equivalent to having the
eigenfunctions of the limiting continuous operator of the graph
laplacian in a separation of variables form. Second, the first
non-trivial Neumann function of any Sturm-Liouville operator is
monotonic. Both the degenerate and non-degenerate spectrums
corresponding to identical and non-identical sources are studied. We
provide successful numerical experiments of the algorithm.