CoSaMP: Iterative signal recovery from incomplete and inaccurate samples

The theory of compressed sensing suggests that successful inversion of an image of the physical world (broadly defined to include speech signals, radar/sonar returns, vibration records, sensor array snapshot vectors, 2-D images, and so on) for its source modes and amplitudes can be achieved at measurement dimensions far lower than what might be expected from the classical theories of spectrum or modal analysis, provided that the image is sparse in an apriori known basis. For imaging problems in spectrum analysis, and passive and active radar/sonar, this basis is usually taken to be a DFT basis. However, in reality no physical field is sparse in the DFT basis or in any apriori known basis. No matter how finely we grid the parameter space the sources may not lie in the center of the grid cells and consequently there is mismatch between the assumed and the actual bases for sparsity. In this paper, we study the sensitivity of compressed sensing to mismatch between the assumed and the actual sparsity bases. We start by analyzing the effect of basis mismatch on the best k-term approximation error, which is central to providing exact sparse recovery guarantees. We establish achievable bounds for the l1 error of the best k -term approximation and show that these bounds grow linearly with the image (or grid) dimension and the mismatch level between the assumed and actual bases for sparsity. We then derive bounds, with similar growth behavior, for the basis pursuit l1 recovery error, indicating that the sparse recovery may suffer large errors in the presence of basis mismatch. Although, we present our results in the context of basis pursuit, our analysis applies to any sparse recovery principle that relies on the accuracy of best k-term approximations for its performance guarantees. We particularly highlight the problematic nature of basis mismatch in Fourier imaging, where spillage from off-grid DFT components turns a sparse representation into an incompressible one. We substantiate our mathematical analysis by numerical examples that demonstrate a considerable performance degradation for image inversion from compressed sensing measurements in the presence of basis mismatch, for problem sizes common to radar and sonar.

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Sensitivity to Basis Mismatch in Compressed Sensing

A non-adapted sparse approximation of PDEs with stochastic inputs

Compressed sensing (CS) has recently emerged as a powerful signal acquisition paradigm. In essence, CS enables the recovery of high-dimensional sparse signals from relatively few linear observations in the form of projections onto a collection of test vectors. Existing results show that if the entries of the test vectors are independent realizations of certain zero-mean random variables, then with high probability the unknown signals can be recovered by solving a tractable convex optimization. This work extends CS theory to settings where the entries of the test vectors exhibit structured statistical dependencies. It follows that CS can be effectively utilized in linear, time-invariant system identification problems provided the impulse response of the system is (approximately or exactly) sparse. An immediate application is in wireless multipath channel estimation. It is shown here that time-domain probing of a multipath channel with a random binary sequence, along with utilization of CS reconstruction techniques, can provide significant improvements in estimation accuracy compared to traditional least-squares based linear channel estimation strategies. Abstract extensions of the main results are also discussed, where the theory of equitable graph coloring is employed to establish the utility of CS in settings where the test vectors exhibit more general statistical dependencies.

/pdf/toeplitz-compressed-sensing-matrices-with-applications-to-3m3324v0d1.pdf

Toeplitz Compressed Sensing Matrices With Applications to Sparse Channel Estimation

Compressed sensing theory suggests that successful inversion of an image of the physical world from its modal parameters can be achieved at measurement dimensions far lower than the image dimension, provided that the image is sparse in an a priori known basis The assumed basis for sparsity typically corresponds to a gridding of the parameter space, eg, an DFT grid in spectrum analysis However, in reality no physical field is sparse in the DFT basis or in an a priori known basis No matter how finely we grid the parameter space the sources may not lie in the center of the grid cells and there is always mismatch between the assumed and the actual bases for sparsity In this paper, we study the sensitivity of compressed sensing (basis pursuit to be exact) to mismatch between the assumed and the actual sparsity bases Our mathematical analysis and numerical examples show that the performance of basis pursuit degrades considerably in the presence of basis mismatch

/pdf/sensitivity-to-basis-mismatch-in-compressed-sensing-2l8u3zva29.pdf

Sensitivity to basis mismatch in compressed sensing

We demonstrate a simple greedy algorithm that can reliably recover a vector v ∈ R from incomplete and inaccurate measurements x = Φv + e. Here Φ is a N × d measurement matrix with N ≪ d, and e is an error vector. Our algorithm, Regularized Orthogonal Matching Pursuit (ROMP), seeks to close the gap between two major approaches to sparse recovery. It combines the speed and ease of implementation of the greedy methods with the strong guarantees of the convex programming methods. For any measurement matrix Φ that satisfies a Uniform Uncertainty Principle, ROMP recovers a signal v with O(n) nonzeros from its inaccurate measurements x in at most n iterations, where each iteration amounts to solving a Least Squares Problem. The noise level of the recovery is proportional to √ logn‖e‖2. In particular, if the error term e vanishes the reconstruction is exact. This stability result extends naturally to the very accurate recovery of approximately sparse signals.

/pdf/signal-recovery-from-inaccurate-and-incomplete-measurements-2lzx63ni6p.pdf

Roman Vershynin

Papers

Signal Recovery from Inaccurate and Incomplete Measurements via Regularized Orthogonal Matching Pursuit