A Structured Construction of Optimal Measurement Matrix for Noiseless Compressed Sensing via Analog Polarization

 Abstract—In this paper, we propose a method of structured construction of the optimal measurement matrix for noiseless compressed sensing (CS), which achieves the minimum number of measurements which only needs to be as large as the sparsity of the signal itself to be recovered to guarantee almost error-free recovery, for sufficiently large dimension. To arrive at the results, we employ a duality between noiseless CS and analog coding across sparse additive noisy channel (SANC). Extending Rényi’s Information Dimension to Mutual Information Dimension (MID), we show the operational meaning of MID to be the fundamental limit of asymptotically error-free analog transmission across SANC under linear analog encoding constraint. We prove that MID polarizes after analog polar transformation and obeys the same recursive relationship as BEC. We further prove that analog polar encoding can achieve the fundamental limit of achievable dimension rate with vanishing Pe across SANC. From the duality, a structured construction scheme is proposed for the linear measurement matrix which achieves the minimum measurement requirement for noiseless CS.