An under-determined system of linear equation has infinitely many solutions (if it has a solution). But, if we have some prior knowledge saying that the solution vector is sparse enough, then under certain conditions there exists a unique solution. Moreover we can find the solution in polynomial time by solving a linear program. This is a celebrated result (by Candes and Tao) that has created a whole new area called *Compressed sensing* . In this talk I will introduce the problem, and will prove a simple version of such reconstruction of sparse vectors from an under-determined system of linear equations.
(For everything that you want to know about compressed sensing, visit http://dsp.rice.edu/cs)