50 research outputs found
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
Recovery under Side Constraints
This paper addresses sparse signal reconstruction under various types of
structural side constraints with applications in multi-antenna systems. Side
constraints may result from prior information on the measurement system and the
sparse signal structure. They may involve the structure of the sensing matrix,
the structure of the non-zero support values, the temporal structure of the
sparse representationvector, and the nonlinear measurement structure. First, we
demonstrate how a priori information in form of structural side constraints
influence recovery guarantees (null space properties) using L1-minimization.
Furthermore, for constant modulus signals, signals with row-, block- and
rank-sparsity, as well as non-circular signals, we illustrate how structural
prior information can be used to devise efficient algorithms with improved
recovery performance and reduced computational complexity. Finally, we address
the measurement system design for linear and nonlinear measurements of sparse
signals. Moreover, we discuss the linear mixing matrix design based on
coherence minimization. Then we extend our focus to nonlinear measurement
systems where we design parallel optimization algorithms to efficiently compute
stationary points in the sparse phase retrieval problem with and without
dictionary learning
Sparsity Promoting Off-grid Methods with Applications in Direction Finding
University of Minnesota Ph.D. dissertation. May 2017. Major: Electrical/Computer Engineering. Advisor: Mostafa Kaveh. 1 computer file (PDF); x, 99 pages.In this dissertation, the problem of directions-of-arrival (DoA) estimation is studied by the compressed sensing application of sparsity-promoting regularization techniques. Compressed sensing can recover high-dimensional signals with a sparse representation from very few linear measurements by nonlinear optimization. By exploiting the sparse representation for the multiple measurement vectors or the spatial covariance matrix of correlated or uncorrelated sources, the DoA estimation problem can be formulated in the framework of sparse signal recovery with high resolution. There are three main topics covered in this dissertation. These topics are recovery methods for the sparse model with structured perturbations, continuous sparse recovery methods in the super-resolution framework, and the off-grid DoA estimation with array self-calibration. These topics are summarized below. For the first topic, structured perturbation in the sparse model is considered. A major limitation of most methods exploiting sparse spectral models for the purpose of estimating directions-of-arrival stems from the fixed model dictionary that is formed by array response vectors over a discrete search grid of possible directions. In general, the array responses to actual DoAs will most likely not be members of such a dictionary. Thus, the sparse spectral signal model with uncertainty of linearized dictionary parameter mismatch is considered, and the dictionary matrix is reformulated into a multiplication of a fixed base dictionary and a sparse matrix. Based on this sparse model, we propose several convex optimization algorithms. However, we are also concerned with the development of a computationally efficient optimization algorithm for off-grid direction finding using a sparse observation model. With an emphasis on designing efficient algorithms, various sparse problem formulations are considered, such as unconstrained formulation, primal-dual formulation, or conic formulation. But, because of the nature of nondifferentiable objective functions, those problems are still challenging to solve in an efficient way. Thus, the Nesterov smoothing methodology is utilized to reformulate nonsmooth functions into smooth ones, and the accelerated proximal gradient algorithm is adopted to solve the smoothed optimization problem. Convergence analysis is conducted as well. The accuracy and efficiency of smoothed sparse recovery methods are demonstrated for the DoA estimation example. In the second topic, estimation of directions-of-arrival in the spatial covariance model is studied. Unlike the compressed sensing methods which discretize the search domain into possible directions on a grid, the theory of super resolution is applied to estimate DoAs in the continuous domain. We reformulate the spatial spectral covariance model into a multiple measurement vectors (MMV)-like model, and propose a block total variation norm minimization approach, which is the analog of Group Lasso in the super-resolution framework and that promotes the group-sparsity. The DoAs can be estimated by solving its dual problem via semidefinite programming. This gridless recovery approach is verified by simulation results for both uncorrelated and correlated source signals. In the last topic, we consider the array calibration issue for DoA estimation, and extend the previously considered single measurement vector model to multiple measurement vectors. By exploiting multiple measurement snapshots, a modified nuclear norm minimization problem is proposed to recover a low-rank matrix with high probability. The definition of linear operator for the MMV model is given, and its corresponding matrix representation is derived so that a reformulated convex optimization problem can be solved numerically. In order to alleviate computational complexity of the method, we use singular value decomposition (SVD) to reduce the problem size. Furthermore, the structured perturbation in the sparse array self-calibration estimation problem is considered as well. The performance and efficiency of the proposed methods are demonstrated by numerical results
Multi-Antenna Dual-Blind Deconvolution for Joint Radar-Communications via SoMAN Minimization
Joint radar-communications (JRC) has emerged as a promising technology for
efficiently using the limited electromagnetic spectrum. In JRC applications
such as secure military receivers, often the radar and communications signals
are overlaid in the received signal. In these passive listening outposts, the
signals and channels of both radar and communications are unknown to the
receiver. The ill-posed problem of recovering all signal and channel parameters
from the overlaid signal is terms as dual-blind deconvolution (DBD). In this
work, we investigate a more challenging version of DBD with a multi-antenna
receiver. We model the radar and communications channels with a few (sparse)
continuous-valued parameters such as time delays, Doppler velocities, and
directions-of-arrival (DoAs). To solve this highly ill-posed DBD, we propose to
minimize the sum of multivariate atomic norms (SoMAN) that depends on the
unknown parameters. To this end, we devise an exact semidefinite program using
theories of positive hyperoctant trigonometric polynomials (PhTP). Our
theoretical analyses show that the minimum number of samples and antennas
required for perfect recovery is logarithmically dependent on the maximum of
the number of radar targets and communications paths rather than their sum. We
show that our approach is easily generalized to include several practical
issues such as gain/phase errors and additive noise. Numerical experiments show
the exact parameter recovery for different JRCComment: 40 pages, 6 figures. arXiv admin note: text overlap with
arXiv:2208.0438