Extracting randomness and dependencies via a matrix polarization

The basic polarization phenomenon for i.i.d. sources is extended to a framework allowing dependencies within and between multiple sources. In particular, it is shown that taking the polar transform of a random matrix with i.i.d. columns of arbitrary (correlated) distribution allows to extract the randomness and dependencies. This result is then used to develop polar coding schemes (having low complexity) for: (1) distributed data compression, i.e., Slepian-Wolf coding (without decomposing the problem into single-user problems), (2) compression of sources with memory, (3) compression of sources on finite fields, extending the polarization phenomenon for alphabets of prime cardinality to powers of primes. A connection to problem of constructing extractors in computer science is discussed.