We present a framework for networked state estimation, where systems encode their (possibly high dimensional) state vectors using a mutually agreed basis between the system and the estimator (in a remote monitoring unit). The basis sparsifies the state vectors, i.e., it represents them using vectors with few non-zero components, and as a result, the systems might need to transmit only a fraction of the original information to be able to recover the non-zero components of the transformed state vector. Hence, the estimator can recover the state vector of the system from an under-determined linear set of equations. We use a greedy search algorithm to calculate the sparsifying basis. Then, we present an upper bound for the estimation error. Finally, we demonstrate the results on a numerical example.