Sudocodes – Fast Measurement and Reconstruction of Sparse Signals

Sudocodes are a new scheme for lossless compressive sampling and reconstruction of sparse signals. Consider a sparse signal x ∈ R containing only K N non-zero values. Sudo-encoding computes the codeword y ∈ R via the linear matrix-vector multiplication y = Φx, with K < M N . We propose a non-adaptive construction of a sparse Φ comprising only the values 0 and 1; hence the computation of y involves only sums of subsets of the elements of x. An accompanying sudodecoding strategy efficiently recovers x given y. Sudocodes require only M = O(K log(N)) measurements for exact reconstruction with worst-case computational complexity O(K log(K) log(N)). Sudocodes can be used as erasure codes for real-valued data and have potential applications in peer-to-peer networks and distributed data storage systems. They are also easily extended to signals that are sparse in arbitrary bases.