Over-the-Air Statistical Estimation of Sparse Models

We propose schemes for minimax statistical estimation of sparse parameter or observation vectors over a Gaussian multiple-access channel (MAC) under squared error loss, using techniques from statistics, compressed sensing and wireless communication. These “analog” schemes exploit the superposition inherent in the Gaussian MAC, using compressed sensing to reduce the number of channel uses needed. For the sparse Gaussian location and sparse product Bernoulli models, we derive expressions for risk in terms of the numbers of nodes, parameters, channel uses and nonzero entries (sparsity). We show that they offer exponential improvements over existing lower bounds for risk in “digital” schemes that assume nodes to transmit bits errorlessly at the Shannon capacity. This shows that analog schemes that design estimation and communication jointly can efficiently exploit the inherent sparsity in high-dimensional models and observations, and provide drastic improvements over digital schemes that separate source and channel coding in this context.