Sparse Recovery Analysis of Preconditioned Frames via Convex Optimization

Orthogonal Matching Pursuit and Basis Pursuit are popular reconstruction algorithms for recovery of sparse signals. The exact recovery property of both the methods has a relation with the coherence of the underlying redundant dictionary, i.e. a frame. A frame with low coherence provides better guarantees for exact recovery. An equivalent formulation of the associated linear system is obtained via premultiplication by a non-singular matrix. In view of bounds that guarantee sparse recovery, it is very useful to generate the preconditioner in such way that the preconditioned frame has low coherence as compared to the original. In this paper, we discuss the impact of preconditioning on sparse recovery. Further, we formulate a convex optimization problem for designing the preconditioner that yields a frame with improved coherence. In addition to reducing coherence, we focus on designing well conditioned frames and numerically study the relationship between the condition number of the preconditioner and the coherence of the new frame. Alongside theoretical justifications, we demonstrate through simulations the efficacy of the preconditioner in reducing coherence as well as recovering sparse signals.Comment: 9 pages, 5 Figure

Optimized Compressed Sensing Matrix Design for Noisy Communication Channels

We investigate a power-constrained sensing matrix design problem for a compressed sensing framework. We adopt a mean square error (MSE) performance criterion for sparse source reconstruction in a system where the source-to-sensor channel and the sensor-to-decoder communication channel are noisy. Our proposed sensing matrix design procedure relies upon minimizing a lower-bound on the MSE. Under certain conditions, we derive closed-form solutions to the optimization problem. Through numerical experiments, by applying practical sparse reconstruction algorithms, we show the strength of the proposed scheme by comparing it with other relevant methods. We discuss the computational complexity of our design method, and develop an equivalent stochastic optimization method to the problem of interest that can be solved approximately with a significantly less computational burden. We illustrate that the low-complexity method still outperforms the popular competing methods.Comment: Submitted to IEEE ICC 2015 (EXTENDED VERSION

Power-Constrained Sparse Gaussian Linear Dimensionality Reduction over Noisy Channels

In this paper, we investigate power-constrained sensing matrix design in a sparse Gaussian linear dimensionality reduction framework. Our study is carried out in a single--terminal setup as well as in a multi--terminal setup consisting of orthogonal or coherent multiple access channels (MAC). We adopt the mean square error (MSE) performance criterion for sparse source reconstruction in a system where source-to-sensor channel(s) and sensor-to-decoder communication channel(s) are noisy. Our proposed sensing matrix design procedure relies upon minimizing a lower-bound on the MSE in single-- and multiple--terminal setups. We propose a three-stage sensing matrix optimization scheme that combines semi-definite relaxation (SDR) programming, a low-rank approximation problem and power-rescaling. Under certain conditions, we derive closed-form solutions to the proposed optimization procedure. Through numerical experiments, by applying practical sparse reconstruction algorithms, we show the superiority of the proposed scheme by comparing it with other relevant methods. This performance improvement is achieved at the price of higher computational complexity. Hence, in order to address the complexity burden, we present an equivalent stochastic optimization method to the problem of interest that can be solved approximately, while still providing a superior performance over the popular methods.Comment: Accepted for publication in IEEE Transactions on Signal Processing (16 pages