Adaptivity provably helps: Information-theoretic limits on l0 cost of non-adaptive sensing

The advantages of adaptivity and feedback are of immense interest in signal processing and communication with many positive and negative results. Although it is established that adaptivity does not offer substantial reductions in minimax mean square error for a fixed number of measurements, existing results have shown several advantages of adaptivity in complexity of reconstruction, accuracy of support detection, and gain in signal-to-noise ratio, under constraints on sensing energy. Sensing energy has often been measured in terms of the Frobenius Norm of the sensing matrix. This paper uses a different metric that we call the l0 cost of a sensing matrix– to quantify the complexity of sensing. Thus sparse sensing matrices have a lower cost. We derive information-theoretic lower bounds on the l0 cost that hold for any non-adaptive sensing strategy. We establish that any nonadaptive sensing strategy must incur an l0 cost of a Θ (N log2(N)) to reconstruct an N -dimensional, one–sparse signal when the number of measurements are limited to Θ (log2(N)). In comparison, bisection-type adaptive strategies only require an l0 cost of at most O(N) for equal order of measurements. The problem has an interesting interpretation as a sphere packing problem in a multidimensional space, such that all the sphere centres have minimum non-zero co-ordinates. We also discuss the variation in l0 cost as the number of measurements increase from Θ (log2(N)) to Θ (N).