Polarization of the Rényi information dimension for single and multi terminal analog compression

This paper shows that the Rényi information dimension (RID) of an i.i.d. sequence of mixture random variables polarizes to the extremal values of 0 and 1 (fully discrete and continuous distributions) when transformed by an Hadamard matrix. This provides a natural counter-part over the reals of the entropy polarization phenomenon over finite fields. It is further shown that the polarization pattern of the RID is equivalent to the BEC polarization pattern, which admits a closed form expression. These results are used to construct universal and deterministic partial Hadamard matrices for analog to analog (A2A) compression of memoryless sources. In addition, a framework for the A2A compression of multi-terminal correlated sources is developed, providing a first counter-part of the Slepian-Wolf coding problem in the A2A setting.