Making Do with Less: An Introduction to Compressed Sensing

This article offers an accessible but rigorous and essentially self-contained account of some of the central ideas in compressed sensing, aimed at nonspecialists and undergraduates who have had linear algebra and some probability. The basic premise is first illustrated by considering the problem of detecting a few defective items in a large set. We then build up the mathematical framework of compressed sensing to show how combining efficient sampling methods with elementary ideas from linear algebra and a bit of approximation theory, optimization, and probability allows the estimation of unknown quantities with far less sampling of data than traditional methods.