Data analysis is fundamental to many parts of modern society. This work examines how we can use sparsity to extract meaning and structure from complex data sets.
We begin by establishing a general framework for imposing a model on the sparse representation of a signal. This not only encompasses many of the existing theoretical results, but allows us to use more complex notions of sparsity in our models. We present both deterministic recovery thresholds and probabilistic results stating when we can recover sparse signals and what we gain by using these more complex notions of sparsity.
Furthermore, we develop a new notion of sparsity which we use to identify peptides in biological samples. We show how we use our new framework to model these particular signals and then, due to the overwhelming problem size, we develop specific algorithms to solve this problem in an efficient manner.