Abstract:

Many statistical techniques can be described as transformations on random vectors which guarantee desired statistical properties of the transformed vectors. Principal component analysis (PCA) is a linear transformation which sequentially maximizes the variances of the components of the transformed vector and also makes them uncorrelated. Linear independent component analysis (ICA) defines a linear transformation that makes the components of the transformed vector mutually independent under certain assumptions. ICA often yields a clearly more meaningful representation of the data than PCA. However, it is not generally possible to make the components mutually independent using a linear transformation. In this paper it is shown that a rectangular self-organizing map approximately defines a ``nonlinear ICA'', a nonlinear smooth mapping that makes the transformed vector components mutually independent. This is justified by showing that a rectangular SOM has approximately uniform distribution on the map.