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 paper 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.