A study of the universal threshold in the ℓ1 recovery by statistical mechanics

We discuss the universality of the ℓ₁ recovery threshold in compressed sensing. Previous studies in the fields of statistical mechanics and random matrix integration have shown that ℓ₁ recovery under a random matrix with orthogonal symmetry has a universal threshold. This indicates that the threshold of ℓ₁ recovery under a non-orthogonal random matrix differs from the universal one. Taking this into account, we use a simple random matrix without orthogonal symmetry, where the random entries are not independent, and show analytically that the threshold of ℓ₁ recovery for such a matrix does not coincide with the universal one. The results of an extensive numerical experiment are in good agreement with the analytical results, which validates our methodology. Though our analysis is based on replica heuristics in statistical mechanics and is not rigorous, the findings nevertheless support the fact that the universality of the threshold is strongly related to the symmetry of the random matrix.