Sparsity-Cognizant Total Least-Squares for Perturbed Compressive Sampling

By Hao Zhu, Geert Leus, Georgios B. Giannakis. IEEE Transactions on Signal Processing, 2010

427 citations recorded by the source corpus.

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 (regularized) 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.

Subfields: Computational Mechanics, Computer Vision and Pattern Recognition, Signal Processing

View Paper: https://www.semanticscholar.org/paper/205595bdc850718e5dae04d283bcb48a4080386f

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Citations in the corpus

Robust Sparse Regression under Adversarial Corruption (2013, methodology): “When E is known to be row sparse (as is the case in our row-corruption model), one might choose ‖·‖∗ to be ‖·‖1,2 or ‖·‖1,∞1; the work in (Zhu et al., 2011) considers using ‖ · ‖∗ = ‖ · ‖F (similar to TLS), which is more suitable when E is dense yet bounded.”
Robust High Dimensional Sparse Regression and Matching Pursuit (2013, methodology): “One might modify this approach in the spirit of Total Least Squares (TLS) [36] to account for noise in the covariates (discussed in Section III), but it leads to highly non convex problems.”
Sparse Methods for Direction-of-Arrival Estimation (2016, methodology): “In [98], a sparse total least-squares (STLS) approach was proposed which, in the single snapshot case, solves the following LASSO-like problem:”
Compressive Sensing in Electromagnetics - A Review (2015, methodology): “In [66], the sparse total least square (S-TLS) method has been proposed to solve the following single-snapshot problem:”
Numerical construction of Green's functions in high dimensional elliptic problems with variable coefficients and analysis of renewable energy data via sparse and separable approximations (2012, methodology): “An algorithm that hybridizes total least squares [59], a computational error in variables model, and the lasso was also recently published [153].”
Off-Grid Direction of Arrival Estimation Using Sparse Bayesian Inference (2011, methodology): “The off-grid distance (the distance from the true DOA to the nearest grid point) that lies in a bounded interval is assumed to be uniformly distributed (noninformative) rather than Gaussian as in [17].”