Various thresholds for $\ell_1$-optimization in compressed sensing (0907.3666v1)

Abstract: Recently, \cite{CRT,DonohoPol} theoretically analyzed the success of a polynomial $\ell_1$-optimization algorithm in solving an under-determined system of linear equations. In a large dimensional and statistical context \cite{CRT,DonohoPol} proved that if the number of equations (measurements in the compressed sensing terminology) in the system is proportional to the length of the unknown vector then there is a sparsity (number of non-zero elements of the unknown vector) also proportional to the length of the unknown vector such that $\ell_1$-optimization succeeds in solving the system. In this paper, we provide an alternative performance analysis of $\ell_1$-optimization and obtain the proportionality constants that in certain cases match or improve on the best currently known ones from \cite{DonohoPol,DT}.

• The paper introduces strong, weak, and sectional threshold analyses for ℓ1-optimization, enhancing recovery guarantees for sparse signals.
• It applies Gordon's Escape through a Mesh theorem to link the null-space properties of random matrices with high-dimensional polytope geometry.
• The study provides detailed numerical threshold estimates that inform the design of effective sensing matrices in compressed sensing applications.

Analysis of Various Thresholds for $\ell_1$-Optimization in Compressed Sensing

The paper, authored by Mihailo Stojnic, presents a comprehensive exploration of $\ell_1$-optimization in the context of compressed sensing, specifically examining various thresholds within the field of under-determined systems of linear equations. This scholarly work offers an alternative performance analysis that aims to provide refined estimates for the success of $\ell_1$-optimization, complementing the foundational studies by Donoho, Candès, Romberg, and Tao.

Key Contributions

The paper explores the mathematical intricacies of $\ell_1$-optimization, seeking to recover sparse signals from fewer measurements than traditionally required by the Nyquist sampling theorem. The principal contributions of this paper are:

1. Strong, Weak, and Sectional Thresholds: The paper introduces a detailed probabilistic analysis to determine these thresholds for $\ell_1$-recovery, showcasing improvements and matches with existing results. The strong threshold addresses recoverability for all sparse signals, while the weak threshold relaxes this to "almost all" signals, and the sectional threshold pertains to signals with a fixed pattern of non-zero entries.
2. Theoretical Framework: Stojnic employs Gordon's "Escape through a Mesh" theorem to provide probabilistic assurances for the success of $\ell_1$-optimization, associating the null-space properties of the measurement matrix with the geometry of high-dimensional polytopes.
3. Implications on Random Matrices and the RIP: The paper suggests that certain random matrices, which have their null-spaces uniformly distributed, could satisfy recovery conditions with overwhelming probability. Stojnic's approach thereby circumvents the stringent demands of the Restricted Isometry Property (RIP), albeit noting that RIP remains a sufficient yet not necessary condition.
4. Impact on Signed and Unsigned Signals: Stojnic extends the analysis to scenarios where signals have predetermined non-negative components, providing theoretical assurances under slightly modified $\ell_1$-optimization models.

Detailed Numerical Results

Stojnic numerically derives threshold values that, in some cases, align or improve upon the prevalent results in the literature. Notably, the threshold values for strong recovery in nearly full-rank matrices approach optimality by matching results from previous studies. For weak thresholds, the results concord with existing benchmarks, demonstrating robustness across a spectrum of sparsity levels and measurement rates.

Implications and Future Directions

The theoretical insights presented have notable implications for the design of sensing matrices and algorithms in compressed sensing. By elucidating conditions under which $\ell_1$-optimization is successful, practitioners can judiciously select measurement matrices in practical applications like signal processing, machine learning, imaging, and more.

Stojnic's analysis invites further exploration into:

• Generalization to $\ell_q$-optimization for $0 < q < 1$, which could reveal more nuanced behaviors of recovery thresholds.
• Investigations into noise-robustness and approximate signal recovery, considering that practical scenarios often deal with noisy measurements and non-ideal sparsity.
• The potential for universal strategies in designing sensing systems informed by the geometric properties of the associated polytopes.

The scholarly rigor and methodological contributions of this paper enrich the ongoing dialogue in the field of compressed sensing, providing a valuable reference point for future research endeavors in high-dimensional data recovery.