Fuzzy Subspace Clustering Algorithm and Applications to Blind Signal Separation

We define a fuzzy subspace skeleton of data points and propose an algorithm for finding it. Such a skeleton is connected with data representation: if the data points (represented as columns of a given matrix X) belong exactly to this fuzzy skeleton, then under some mild conditions we can represent X of the form X = AS uniquely (up to scaling and permutation), where the matrices A and S with dimensions m times m and m times N respectively (often called mixing matrix or dictionary and source matrix) are such that S is r-sparse in sense that each column of S has at most m - r nonzero elements. In this paper we consider the case r ges 2 and develop a fuzzy algorithm for clustering over subspaces, which is essential for identification of the mixing matrix A. The idea of this clustering is the same as in the classical fuzzy clustering problem, but instead of balls, here we cluster over subspaces with co-dimension r. For identification of the source matrix, we apply a special source recovery algorithm. We illustrate our algorithms with examples. We note that our method is quite general, since the sparseness conditions could be obtained with some preprocessing methods and no independence conditions for the source signals are imposed (in contrast to independent component analysis).