We consider the problem of testing for the presence (or detection) of an unknown sparse signal in additive white noise. Given a fixed measurement budget, much smaller than the dimension of the signal, we consider the general problem of designing compressive measurements to maximize the measurement signal-to-noise ratio (SNR), as increasing SNR improves the detection performance in a large class of detectors. We use a lexicographic optimization approach, where the optimal measurement design for sparsity level k is sought only among the set of measurement matrices that satisfy the optimality conditions for sparsity level k−1. We consider optimizing two different SNR criteria, namely a worst-case SNR measure, over all possible realizations of a k-sparse signal, and an average SNR measure with respect to a uniform distribution on the locations of the up to k nonzero entries in the signal. We establish connections between these two criteria and certain classes of tight frames. We constrain our measurement matrices to the class of tight frames to avoid coloring the noise covariance matrix. For the worst-case problem, we show that the optimal measurement matrix is a Grassmannian line packing for most-and a uniform tight frame for allsparse signals. For the average SNR problem, we prove that the optimal measurement matrix is a uniform tight frame with minimum sum-coherence for most-and a tight frame for all-sparse signals.

Detecting a sparse signal in noise is fundamentally different from reconstructing a sparse signal, as the objective is to optimize a detection performance criterion rather than to find the sparsest signal that satisfies a linear observation equation. In this paper, we consider the design of lowdimensional (compressive) measurement matrices for detecting sparse signals in white Gaussian noise. We use a lexicographic optimization approach to maximize the worst-case signal-tonoise ratio (SNR). More specifically, we find an optimal solution for a k-sparse signal among optimal solutions subject to sparsity level k − 1. We show that for all sparse signals, columns of the optimal measurement matrix must form a uniform tight frame. For 2-sparse signals, the smallest angle among angles between element pairs of this frame must be maximized. In this case, the optimal solution matrix is an optimal Grassmannian packing. For k-sparse signals where k > 2, the largest angle among such angles must be as close to the maximum smallest angle as possible. We show that under certain conditions, columns of the optimal measurement matrix form an equiangular uniform tight frame. For this case, we derive an expression for the maximal SNR in the worst-case scenario, as a function of the signal dimension and the number of measurements.

We consider the problem of adaptively designing compressive measurement matrices for estimating time-varying sparse signals. We formulate this problem as a partially observable Markov decision process. This formulation allows us to use Bellman's principle of optimality in the implementation of multistep lookahead designs of compressive measurements. We compare the performance of adaptive versus traditional non-adaptive designs and study the value of multi-step (non-myopic) versus one-step (myopic) lookahead adaptive schemes by introducing two variations of the compressive measurement design problem. In the first variation, we consider the problem of sequentially selecting measurement matrices with fixed dimensions from a prespecified library of measurement matrices. In the second variation, the number of compressive measurements, i.e., the number of rows of the measurement matrix, is adaptively chosen. Once the number of measurements is determined, the matrix entries are chosen according to a prespecified adaptive scheme. Each of these two problems is judged by a separate performance criterion. The gauge of efficiency in the first problem is the conditional mutual information between the sparse signal support and measurements. The second problem applies a linear combination of the number of measurements and conditional mutual information as the performance measure. Through several simulations, we study the effectiveness of different designs in various settings. The primary focus in these simulations is the application of a method known as rollout. However, the computational load required for using the rollout method has also inspired us to adapt two data association heuristics to the compressive sensing paradigm. These heuristics show promising decreases in the amount of computation for propagating distributions and searching for optimal solutions.

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We present an approach to adaptive measurement selection in compressive sensing for estimating sparse signals. Given a fixed number of measurements, we consider the sequential selection of the rows of a compressive measurement matrix to maximize the mutual information between the measurements and the sparse signal's support. We formulate this problem as a partially observable Markov decision process (POMDP), which enables the application of principled reasoning for sequential measurement selection based on Bellman's optimality condition.