Information-theoretically optimal compressed sensing via spatial coupling and approximate message passing

We study the compressed sensing reconstruction problem for a broad class of random, band-diagonal sensing matrices. This construction is inspired by the idea of spatial coupling in coding theory. As demonstrated heuristically and numerically by Krzakala et al. [11], message passing algorithms can effectively solve the reconstruction problem for spatially coupled measurements with undersampling rates close to the fraction of non-zero coordinates. We use an approximate message passing (AMP) algorithm and analyze it through the state evolution method. We give a rigorous proof that this approach is successful as soon as the undersampling rate δ exceeds the (upper) Rényi information dimension of the signal, d̅(p
_X
). More precisely, for a sequence of signals of diverging dimension n whose empirical distribution converges to p
_X
, reconstruction is with high probability successful from d̅(p
_X
)n+o(n) measurements taken according to a band diagonal matrix. For sparse signals, i.e. sequences of dimension n and k(n) non-zero entries, this implies reconstruction from k(n) + o(n) measurements. For `discrete' signals, i.e. signals whose coordinates take a fixed finite set of values, this implies reconstruction from o(n) measurements. The result is robust with respect to noise, does not apply uniquely to random signals, but requires the knowledge of the empirical distribution of the signal p
_X
.