ICA over Finite Fields

Independent Component Analysis is usually performed over the fields of reals or complex numbers and the only other field where some insight has been gained so far is GF(2), the finite field with two elements. We extend this to arbitrary finite fields, proving separability of the model if the sources are non-uniform and non-degenerate and present algorithms performing this task. Usually, Independent Component Analysis (ICA) is performed over R, the field of reals, and some extensions of this to the complex case have been performed [1, 3]. Other than this, the only other case that has been investigated, is the case of GF(2), the field with two elements [7]. However this is not the only case of a finite field, rather generally for any prime power q = p there is a finite field with q elements, denoted GF(q). These fields most prominently find application in coding theory, e.g. with low-density parity-check (LDPC) codes, and although here GF(2) finds the most attention, other finite fields also are of interest too, for instance see [5] for a statistical physics based analysis of LDPC codes over finite fields. We present a separation theorem for ICA over arbitrary finite fields, as long as the sources are all non-uniform and have nowhere probability mass 0, generalizing the results in [7], suggest algorithms to efficiently solve the ICA task and show simulations showing the validity of the approach. 1 Finite fields and discrete probability transformations