General Deviants: An Analysis of Perturbations in Compressed Sensing (0907.2955v1)

Abstract: We analyze the Basis Pursuit recovery of signals with general perturbations. Previous studies have only considered partially perturbed observations Ax + e. Here, x is a signal which we wish to recover, A is a full-rank matrix with more columns than rows, and e is simple additive noise. Our model also incorporates perturbations E to the matrix A which result in multiplicative noise. This completely perturbed framework extends the prior work of Candes, Romberg and Tao on stable signal recovery from incomplete and inaccurate measurements. Our results show that, under suitable conditions, the stability of the recovered signal is limited by the noise level in the observation. Moreover, this accuracy is within a constant multiple of the best-case reconstruction using the technique of least squares. In the absence of additive noise numerical simulations essentially confirm that this error is a linear function of the relative perturbation.

• The paper introduces a fully perturbed compressed sensing model that accounts for both additive and multiplicative noise.
• It extends RIP analysis to derive conditions ensuring stable signal recovery despite matrix perturbations.
• Numerical simulations demonstrate that BP reconstruction errors scale linearly with noise levels, guiding robust application.

Analysis of Perturbations in Compressed Sensing

The paper "General Deviants: An Analysis of Perturbations in Compressed Sensing" by Matthew A. Herman and Thomas Strohmer provides a rigorous exploration of signal recovery within the framework of compressed sensing (CS) when both additive and multiplicative noise perturbations are introduced. Historically, CS research has primarily focused on signal recovery under scenarios of sparsity with unperturbed or solely additively perturbed measurement matrices. This paper is notable for extending this exploration to account for perturbations affecting the measurement matrix itself, effectively characterizing them as multiplicative noise.

Main Contribution

The authors introduce a "completely perturbed" model, where they allow the sensing matrix $A$ to be perturbed into a form $A + E$ with $E$ representing perturbations. The paper extends prior results established by Candès, Romberg, and Tao on stability in signal recovery, providing theoretical frameworks and conditions under which Basis Pursuit (BP) can successfully recover signals amidst both additive and multiplicative noise environments. The central contribution is the derivation of conditions under which the stability of recovered signals remains limited by the noise level in observations. More concretely, they demonstrate that this stability can be bounded by a factor of the best-case reconstruction achievable via least squares, given the noise constraints.

Theoretical Insights

Two primary theorems form the backbone of the paper's contributions:

1. RIP Extensions: The authors derived a generalized Restricted Isometry Property (RIP) for perturbed sensing matrices $A + E$. This extension shows that the penalty on the spectrum of the matrix, when submatrices are considered, scales as a linear function of the imposed perturbation level. This is an important addition as it allows for more generalized analysis involving perturbations and provides a mathematical bounding for the RICs of the perturbed matrices.
2. Stability Analysis: The impact of perturbations on BP recovery was quantitatively characterized. Under specific conditions concerning relative perturbation estimates and signal sparsity, the stability of the recovered signal was related directly to the perturbations. A detailed analysis demonstrated that the perturbation in the signal solution scales linearly with perturbation to the measurement matrix, leading to accurate predictions regarding signal recovery fidelity.

Numerical Validation

The numerical simulations presented corroborate the theoretical findings. The results marked by trials varying the perturbation terms revealed that the relative error in the BP solutions scales linearly with the multiplicative noise applied. The simulations emphasized the critical dependency of observation stability on noise levels, affirming that suppressed perturbation values directly correlate with stabilized BP outcomes.

Implications and Future Directions

The implications of this work lie in its potential applications across numerous fields involving CS, including remote sensing, radar, and communication signal processing. Given the increasing reliance on CS in these fields, extending the foundational theory to account for entirety-perturbed systems could enhance the robustness of signal processing systems used in practical applications.

Future research can explore deeper into structured perturbations and the impact of different classes of perturbations beyond the typical Gaussian or Bernoulli distribution models. Moreover, from a methodological standpoint, advancing techniques that allow for efficient computations of RICs in real-time scenarios with perturbations could significantly benefit practical implementations in more dynamic environments.

In conclusion, the paper by Herman and Strohmer enriches the theoretical landscape of CS by addressing a less-explored facet—the inclusion of matrix perturbations—and provides strong mathematical backing alongside empirical validation to support their claims. This elevates the resilience and applicability of CS in various high-noise settings, thereby extending its utility and reliability in advanced signal processing endeavors.