Fixed point theory and semidefinite programming for computable performance analysis of block-sparsity recovery
This work provides a practical tool for analyzing block-sparsity recovery in applications like sensor arrays and radar, though the approach is incremental as it extends existing fixed-point and SDP techniques to a new domain.
The paper develops computable performance bounds for convex block-sparsity recovery algorithms using fixed point theory and semidefinite programming, enabling optimal sensing matrix design. The bounds are shown to be nontrivial for random matrices when the number of measurements is sufficiently large.
In this paper, we employ fixed point theory and semidefinite programming to compute the performance bounds on convex block-sparsity recovery algorithms. As a prerequisite for optimal sensing matrix design, a computable performance bound would open doors for wide applications in sensor arrays, radar, DNA microarrays, and many other areas where block-sparsity arises naturally. We define a family of goodness measures for arbitrary sensing matrices as the optimal values of certain optimization problems. The reconstruction errors of convex recovery algorithms are bounded in terms of these goodness measures. We demonstrate that as long as the number of measurements is relatively large, these goodness measures are bounded away from zero for a large class of random sensing matrices, a result parallel to the probabilistic analysis of the block restricted isometry property. As the primary contribution of this work, we associate the goodness measures with the fixed points of functions defined by a series of semidefinite programs. This relation with fixed point theory yields efficient algorithms with global convergence guarantees to compute the goodness measures.