Lecture 5 1 Overview 2 Compressed Sensing

In the previous lecture, we considered the following sparse approximations problem (w.r.t. Lp norm): For a vector x, find x′ such that x′ is k-sparse, i.e., ‖x‖0 ≤ k, and ‖x− x‖p ≤ C Err, where Err = Errpk = min‖x′′‖0≤k ‖x− x ‖p. We could maintain a linear sketch of x of length O(k logm) from which can recover k-sparse approximation of x (for some C > 1). The guarantee was probabilistic: for any x, the randomly chosen linear sketch worked with “good” probability.