The Limits of Error Correction with lp Decoding

An unknown vector f in R^n can be recovered from corrupted measurements y = Af + e where A^(m*n)(m>n) is the coding matrix if the unknown error vector e is sparse. We investigate the relationship of the fraction of errors and the recovering ability of lp-minimization (0 < p <= 1) which returns a vector x minimizing the "lp-norm" of y - Ax. We give sharp thresholds of the fraction of errors that determine the successful recovery of f. If e is an arbitrary unknown vector, the threshold strictly decreases from 0.5 to 0.239 as p increases from 0 to 1. If e has fixed support and fixed signs on the support, the threshold is 2/3 for all p in (0, 1), while the threshold is 1 for l1-minimization.