Adaptive compressive measurement design using approximate dynamic programming

We consider two problems for adaptive design of compressive measurement matrices for estimating time-varying sparse signals. In the first problem, we fix the number of compressive measurements collected at each time step and design the compressive measurement matrices over time. The goal is to maximize the conditional mutual information between the support of the sparse signal and the measurements. In the second problem, we adaptively select the number of compressive measurements to be taken at each time step and not the entries in the measurement matrices. Once the number of measurements to be taken is determined, the entries are selected according to a prespecified scheme. Here, we optimize a measure that is a combination of the number of measurements and the conditional mutual information between the support of the sparse signal and the measurements at each time step. We formulate both problems as Partially Observable Markov Decision Processes (POMDPs) and use an approximation method known as rollout to find solutions for these problems. The POMDP formulation enables the application of Bellman´s principle for optimality in multi-step lookahead design of compressive measurements.