Sparsity-Cognizant Total Least-Squares for Perturbed Compressive Sampling

Solving linear regression problems based on the total least-squares (TLS) criterion has well-documented merits in various applications, where perturbations appear both in the data vector as well as in the regression matrix. However, existing TLS approaches do not account for sparsity possibly present in the unknown vector of regression coefficients. On the other hand, sparsity is the key attribute exploited by modern compressive sampling and variable selection approaches to linear regression, which include noise in the data, but do not account for perturbations in the regression matrix. The present paper fills this gap by formulating and solving TLS optimization problems under sparsity constraints. Near-optimum and reduced-complexity suboptimum sparse (S-) TLS algorithms are developed to address the perturbed compressive sampling (and the related dictionary learning) challenge, when there is a mismatch between the true and adopted bases over which the unknown vector is sparse. The novel S-TLS schemes also allow for perturbations in the regression matrix of the least-absolute selection and shrinkage selection operator (Lasso), and endow TLS approaches with ability to cope with sparse, under-determined "errors-in-variables" models. Interesting generalizations can further exploit prior knowledge on the perturbations to obtain novel weighted and structured S-TLS solvers. Analysis and simulations demonstrate the practical impact of S-TLS in calibrating the mismatch effects of contemporary grid-based approaches to cognitive radio sensing, and robust direction-of-arrival estimation using antenna arrays.Comment: 30 pages, 10 figures, submitted to IEEE Transactions on Signal Processin

Model-Based Calibration of Filter Imperfections in the Random Demodulator for Compressive Sensing

The random demodulator is a recent compressive sensing architecture providing efficient sub-Nyquist sampling of sparse band-limited signals. The compressive sensing paradigm requires an accurate model of the analog front-end to enable correct signal reconstruction in the digital domain. In practice, hardware devices such as filters deviate from their desired design behavior due to component variations. Existing reconstruction algorithms are sensitive to such deviations, which fall into the more general category of measurement matrix perturbations. This paper proposes a model-based technique that aims to calibrate filter model mismatches to facilitate improved signal reconstruction quality. The mismatch is considered to be an additive error in the discretized impulse response. We identify the error by sampling a known calibrating signal, enabling least-squares estimation of the impulse response error. The error estimate and the known system model are used to calibrate the measurement matrix. Numerical analysis demonstrates the effectiveness of the calibration method even for highly deviating low-pass filter responses. The proposed method performance is also compared to a state of the art method based on discrete Fourier transform trigonometric interpolation.Comment: 10 pages, 8 figures, submitted to IEEE Transactions on Signal Processin

Signal Recovery in Perturbed Fourier Compressed Sensing

In many applications in compressed sensing, the measurement matrix is a Fourier matrix, i.e., it measures the Fourier transform of the underlying signal at some specified `base' frequencies $\{u_i\}_{i=1}^M$, where $M$ is the number of measurements. However due to system calibration errors, the system may measure the Fourier transform at frequencies $\{u_i + \delta_i\}_{i=1}^M$ that are different from the base frequencies and where $\{\delta_i\}_{i=1}^M$ are unknown. Ignoring perturbations of this nature can lead to major errors in signal recovery. In this paper, we present a simple but effective alternating minimization algorithm to recover the perturbations in the frequencies \emph{in situ} with the signal, which we assume is sparse or compressible in some known basis. In many cases, the perturbations $\{\delta_i\}_{i=1}^M$ can be expressed in terms of a small number of unique parameters $P \ll M$. We demonstrate that in such cases, the method leads to excellent quality results that are several times better than baseline algorithms (which are based on existing off-grid methods in the recent literature on direction of arrival (DOA) estimation, modified to suit the computational problem in this paper). Our results are also robust to noise in the measurement values. We also provide theoretical results for (1) the convergence of our algorithm, and (2) the uniqueness of its solution under some restrictions.Comment: New theortical results about uniqueness and convergence now included. More challenging experiments now include

Topics in Compressed Sensing

Compressed sensing has a wide range of applications that include error correction, imaging, radar and many more. Given a sparse signal in a high dimensional space, one wishes to reconstruct that signal accurately and efficiently from a number of linear measurements much less than its actual dimension. Although in theory it is clear that this is possible, the difficulty lies in the construction of algorithms that perform the recovery efficiently, as well as determining which kind of linear measurements allow for the reconstruction. There have been two distinct major approaches to sparse recovery that each present different benefits and shortcomings. The first, L1-minimization methods such as Basis Pursuit, use a linear optimization problem to recover the signal. This method provides strong guarantees and stability, but relies on Linear Programming, whose methods do not yet have strong polynomially bounded runtimes. The second approach uses greedy methods that compute the support of the signal iteratively. These methods are usually much faster than Basis Pursuit, but until recently had not been able to provide the same guarantees. This gap between the two approaches was bridged when we developed and analyzed the greedy algorithm Regularized Orthogonal Matching Pursuit (ROMP). ROMP provides similar guarantees to Basis Pursuit as well as the speed of a greedy algorithm. Our more recent algorithm Compressive Sampling Matching Pursuit (CoSaMP) improves upon these guarantees, and is optimal in every important aspect

Compressive Recovery of Signals Defined on Perturbed Graphs

Recovery of signals with elements defined on the nodes of a graph, from compressive measurements is an important problem, which can arise in various domains such as sensor networks, image reconstruction and group testing. In some scenarios, the graph may not be accurately known, and there may exist a few edge additions or deletions relative to a ground truth graph. Such perturbations, even if small in number, significantly affect the Graph Fourier Transform (GFT). This impedes recovery of signals which may have sparse representations in the GFT bases of the ground truth graph. We present an algorithm which simultaneously recovers the signal from the compressive measurements and also corrects the graph perturbations. We analyze some important theoretical properties of the algorithm. Our approach to correction for graph perturbations is based on model selection techniques such as cross-validation in compressed sensing. We validate our algorithm on signals which have a sparse representation in the GFT bases of many commonly used graphs in the network science literature. An application to compressive image reconstruction is also presented, where graph perturbations are modeled as undesirable graph edges linking pixels with significant intensity difference. In all experiments, our algorithm clearly outperforms baseline techniques which either ignore the perturbations or use first order approximations to the perturbations in the GFT bases.Comment: 18 pages, 15 figures. v2: Minor correction in ref [32