Compressed sensing for finite-valued signals

In this work, we present an approach that incorporates discrete values prior into basis pursuit. We consider bipolar finite-valued and unipolar finite-valued sparse signals, i.e., sparse signals with entries in {−L1,…,L2}, respectively in {0,…,L}, with L1,L2,L∈N. For those signals, we will show that the phase transition for our approach takes place earlier than in the case of basis pursuit. We will in particular derive highly improved performance guarantees for the special type of unipolar binary and bipolar ternary sparse signals, i.e., sparse signals having entries in {0,1}, respectively in {−1,0,1}. More precisely, we will show that independently of the sparsity of the signal, at most N/2, respectively 3N/4, measurements are necessary to recover a unipolar binary, and a bipolar ternary signal uniquely, where N is the dimension of the ambient space. We will further discuss robustness of the algorithm and phase transition under noisy measurements.