Restricted Isometry Property in Quantized Network Coding of sparse messages

In this paper, we study joint network coding and distributed source coding of inter-node dependent messages, with the perspective of compressed sensing. Specifically, the theoretical guarantees for robust ℓ₁-min recovery of an under-determined set of linear network coded sparse messages are investigated. We discuss the guarantees for ℓ₁-min decoding of quantized network coded messages, based on Restricted Isometry Property (RIP) of the resulting measurement matrix. This is done by deriving the relation between tail probability of ℓ₂-norms and satisfaction of RIP. The obtained relation is then used to compare our designed measurement matrix, with i.i.d. Gaussian measurement matrix, in terms of RIP satisfaction. Finally, we present our numerical evaluations, which shows that the proposed design of network coding coefficients results in a measurement matrix with an RIP behavior, similar to that of i.i.d. Gaussian matrix.