PHASE TRANSITIONS IN ERROR CORRECTING AND COMPRESSED SENSING BY ℓ1 LINEAR PROGRAMMING

In correcting a real linear code y = Bx + w by ℓ1 linear programming, where the encoding matrix B ∈ ℝm × n has full rank with m ≥ n and the noise w ∈ ℝm is a sparse random vector, it is numerically observed that the breakdown points of 50% successes in recovering the input vector x ∈ ℝn from the corrupted oversampled measurement y lie on the Donoho–Tanner curves when reflected in their midpoint. The curves of 50% successes in solving underdetermined systems, z = Aw, by ℓ1 linear programming with uniformly distributed compressed sensing matrices A ∈ ℝd × m, where d < m and w is a sparse vector, have been numerically observed and recently shown to coincide with the Donoho–Tanner curves for normally-distributed compressed sensing matrices A derived from geometric combinatorics. When n ≤ m/2, correcting a linear code is faster if done directly by ℓ1 linear programming. However, when n > m/2, to save computing time, this problem can be transformed into an underdetermined compressed sensing problem, Aw = z := Ay, for the syndrome z by a full rank matrix A ∈ ℝd × m, d = m – n, such that AB = 0. For this purpose, to have equivalently high mean breakdown points by ℓ1 linear programming, one can use uniformly distributed random matrices A ∈ ℝ(m-n) × m and matrices B ∈ ℝm × n with orthonormal columns spanning the null space of A. Two exceptional cases have been found. Numerical results are collected in figures and tables.