Adaptive sensing using deterministic partial Hadamard matrices

This paper investigates the construction of determin- istic measurement matrices preserving the entropy of a random vector with a given probability distribution. In particular, it is shown that for a random vector with i.i.d. discrete components, this is achieved by selecting a subset of rows of a Hadamard matrix such that (i) the selection is deterministic (ii) the fraction of selected rows is vanishing. In contrast, it is shown that for a random vector with i.i.d. continuous components, no entropy preserving measurement matrix allows dimensionality reduction. These results are in agreement with the results of Wu-Verdu on almost lossless analog compression. This paper is however motivated by the complexity attribute of Hadamard matrices, which allows the use of efficient and stable reconstruction algo- rithms. The proof technique is based on a polar code martingale argument and on a new entropy power inequality for integer- valued random variables.