Learning algorithms for an information theoretic paradigm of compressive sensing

Abstract

The efficient representation of data in a low dimensional space and its accurate newlinerecovery are essential for its cost-effective storage, processing, and transmission. newlineCompressive sensing comes as a solution to this requirement. In compressive newline newlinesensing, the signal (data) is acquired such that no further compression after ac- newlinequisition of the signal is needed. newline newlineThe crux of compressive sensing lies in the non-uniform sampling of a signal newlineat an average rate much less than the Nyquist rate, and recovering the signal newlineuniquely with a high probability from these reduced set of measurements. The newlinerecovery of the signal from the reduced set of measurements relies on the sparsity newlinestructure of the signal. Hence, the need for identifying an operator that generates newlinea maximally sparse representation of the signal arises. This is the first problem newlineaddressed in this work. newline newlineHaving established the sparsity of a signal with respect to a basis, the im- newlinemediate problem is to identify a sensing operator. The sensing operator should newline newlinebe efficient such as to capture the vital information content of the signal into newlinea reduced set of discrete measurements having cardinality much less than that newlinestipulated by the classical sampling theorem. The fundamental requirement of a newlinesensing operator is its low mutual coherence. Hence, the natural choice for the newlinesensing operator is a random matrix. But structured sensing matrices, proposed newline newlinein the literature, have been proved to outperform the classical choice of the ran- newlinedom matrix as a sensing operator. The second problem addressed in this work is newline newlineto identify efficient sensing operators that do not assume structured sparsity of newlinethe signal. newline