  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.
