## Polarization of the Renyi Information Dimension With Applications to Compressed Sensing

In this paper, we show that the Hadamard matrix acts as an extractor over the reals of the Renyi Information Dimension (RID), in an analogous way to how it acts as an extractor of the discrete entropy over finite fields. More precisely, we prove that the RID of an i.i.d. sequence of mixture random variables polarizes to the extremal values of 0 and 1 (corresponding to discrete and continuous distributions) when transformed by a Hadamard matrix. Furthermore, we prove that the polarization pattern of the RID admits a closed form expression and follows exactly the Binary Erasure Channel (BEC) polarization pattern in the discrete setting. We discuss the applications of the RID polarization to Compressed Sensing of i.i.d. sources. In particular, we use the RID polarization to construct a family of deterministic +/- 1-valued sensing matrices for Compressed Sensing. We run numerical simulations to compare the performance of the resulting matrices with that of the random Gaussian and the random Hadamard matrices. The results indicate that the proposed matrices aftbrd competitive performances, while being explicitly constructed.

Published in:
Ieee Transactions On Information Theory, 63, 11, 6858-6868
Presented at:
IEEE International Symposium on Information Theory (ISIT), Istanbul, TURKEY, JUL 07-12, 2013
Year:
2017
Publisher:
Piscataway, Ieee-Inst Electrical Electronics Engineers Inc
ISSN:
0018-9448
Keywords:
Laboratories: