Sparsity-Cognizant Total Least-Squares for Perturbed Compressive Sampling (1008.2996v1)

Abstract: 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.

• The paper introduces the S-TLS framework by integrating sparsity into TLS to effectively manage matrix perturbations in under-determined models.
• It develops near-optimum, reduced-complexity algorithms that outperform traditional methods like Basis Pursuit and Lasso in sparse signal recovery under low SNR.
• Practical applications in cognitive radio sensing and direction-of-arrival estimation demonstrate the method’s enhanced accuracy and robustness.

Overview of Sparsity-Cognizant Total Least-Squares in Perturbed Compressive Sampling

The paper "Sparsity-Cognizant Total Least-Squares for Perturbed Compressive Sampling," authored by Hao Zhu, Geert Leus, and Georgios B. Giannakis, introduces a novel framework that integrates sparsity considerations into the traditional Total Least-Squares (TLS) approach, specifically for linear regression problems complicated by perturbations. Addressing the inadequacies of existing TLS and compressive sampling methods, this paper expands the theoretical and practical scope of TLS by incorporating sparsity constraints and proposing solutions fit for under-determined models.

Problem Statement and Contributions

The paper identifies a critical gap in the literature where standard TLS approaches fail to consider sparsity in regression coefficient vectors, an attribute vital for modern compressive sampling techniques. Traditional methods like Basis Pursuit and Lasso manage data noise but disregard perturbations in the regression matrix, which can significantly impact performance, particularly in under-determined systems. By formulating Sparse TLS (S-TLS) optimization problems, the authors systematically approach these challenges with developed near-optimum algorithms that reduce complexity effectively.

Key contributions include:

• S-TLS Formulation: This framework offers a robust approach to handle simultaneous sparsity and perturbations in linear models, providing consistent estimators even in under-determined scenarios.
• Algorithm Development: The authors propose near-optimum and reduced-complexity suboptimal S-TLS algorithms aimed at addressing the challenges posed by fully perturbed compressive sampling models.
• Generalizations for Weighted and Structured Perturbations: Expanding S-TLS to exploit known perturbation structures or weightings, paving the way for novel solvers accommodating these conditions.
• Practical Applications: Demonstrated through cognitive radio sensing and direction-of-arrival estimation, the paper illustrates substantial performance improvements in calibrating grid-based mismatches using simulated data.

Strong Numerical Results

Numerical simulations underline the efficacy of S-TLS relative to traditional BP and Lasso approaches. The algorithms' performance reveals remarkable capabilities in reconstructing sparse vectors under perturbed models. Notably, the proposed techniques show impressive robustness and precision in identifying correct signal supports even under low SNR conditions, which are traditionally challenging for conventional methods.

Implications and Future Directions

The inclusion of sparsity into the TLS paradigm not only enhances the robustness of signal reconstruction in noisy environments but also extends the applicability of TLS to a wider range of real-world problems that rely on accurate parameter estimation under uncertainty. The authors’ work prompts the exploration of adaptive, real-time implementations of S-TLS, especially relevant in dynamic systems like cognitive radio networks and sonar systems. Additionally, the integration of structured perturbations further indicates a pathway to tailor S-TLS applications in various other domains where prior structural information is available.

Conclusion

This paper significantly enriches the existing body of knowledge by cross-fertilizing concepts from compressive sensing and total least squares under a unified framework that addresses both sparsity and perturbations. It opens up promising avenues for both theoretical development and practical applications, urging future research to delve into performance analysis and expand these ideas to broader domains within signal processing and statistical learning.