Adaptive Estimation of Time-Varying Sparse Signals

Abstract

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.

Policy Rollout Action Selection in Continuous Domains for Sensor Path Planning

“…If the action space contains a large number of discrete actions, evaluating each might not be feasible. An option to speed up the search is then to prune this space prior to the evaluation, which has been performed in [18], [19].…”

Subspace Selection for Projection Maximization With Matroid Constraints

“…The geometrical interpretation of the above problem is to select K columns of matrix H such that the norm of the projection of η onto the subspace spanned by the chosen columns is maximized. Adaptive algorithms such as those based on partially observable Markov decision processes have been proposed to find the optimal solution [19]. The computation complexity for adaptive algorithms is in general quite high despite the reduction brought by approximation methods such as rollout.…”

Optimal Sparse Output Feedback Control Design: a Rank Constrained Optimization Approach

“…Proof: Let Y * be the optimum of (19), then rank(Y * 11 ) = n and it satisfies both equality and inequality constraints. Therefore, it belongs to the feasible set of (20). Furthermore, for every point Y in the feasible set of (20) with the rank greater than n, we have…”

“…The problem of minimizing the number of nonzero elements of a vector/matrix subject to a set of constraints in inherently NP-hard and arises in many fields, such as Compressive Sensing (CS) where the inherent sparseness of signals is exploited in determining them from relatively few measurements [18]. Since the advent of Compressive Sensing, considerable work has been done on the design of compressive measurement matrices based on different criteria such as sparse signal support detection and estimation [19], [20], sparse signal detection and classification [21], [22], etc.…”