NEW THEORY AND METHODS IN ADAPTIVE AND COMPRESSIVE SAMPLING FOR SPARSE DISCOVERY by Jarvis

The study of sparsity has recently garnered significant attention in the signal processing and statistics communities. Generally speaking, sparsity describes the phenomenon where a large data set may be succinctly represented or approximated using only a small number of summary values or coefficients. The implications are clear—the presence of sparsity suggests the potential for efficient methods to extract only the relevant information, conserving acquisition and/or processing resources which can often be scarce or expensive. The overall theme of this work is the identification of efficient and effective ways to exploit sparsity in a variety of settings and applications. The first part of this work comprises contributions to the emerging theory of compressive sampling (also called compressed sensing or compressive sensing). Compressive sampling describes a framework under which sparse, high-dimensional signals (vectors) can be recovered from a relatively small number of non-adaptive observations. The theory of compressive sampling is extended here, where it is shown that compressive sampling can be an effective tool for sparse recovery in noisy environments, in applications in sensor networking, system identification (channel sensing), and wideband RF surveillance. The second part of this work examines adaptivity in sampling. Adaptive sampling strategies are those which direct subsequent observations based on the results of previous observations, in an effort to focus on features of interest. It is shown here that adaptivity can result in dramatic improvements in recoverability of sparse signals, providing new insight into the fundamental theoretical limits of sparse recovery in noisy settings. A simple adaptive procedure called distilled sensing is proposed and shown to dramatically outperform the best possible non-adaptive strategies, in the sense that distilled sensing enables the recovery (detection and estimation) of signals whose features are otherwise too weak to be recoverable using any methods based on the best non-adaptive sampling strategies. NEW THEORY AND METHODS IN ADAPTIVE AND COMPRESSIVE SAMPLING FOR SPARSE DISCOVERY