CoSaMP: Iterative signal recovery from incomplete and inaccurate samples (0803.2392v2)

Abstract: Compressive sampling offers a new paradigm for acquiring signals that are compressible with respect to an orthonormal basis. The major algorithmic challenge in compressive sampling is to approximate a compressible signal from noisy samples. This paper describes a new iterative recovery algorithm called CoSaMP that delivers the same guarantees as the best optimization-based approaches. Moreover, this algorithm offers rigorous bounds on computational cost and storage. It is likely to be extremely efficient for practical problems because it requires only matrix-vector multiplies with the sampling matrix. For many cases of interest, the running time is just O(N*log^2(N)), where N is the length of the signal.

• The paper introduces CoSaMP, a novel iterative method for accurately recovering sparse signals from compressive and noisy measurements.
• It employs a proxy-based support identification and least-squares estimation process to refine signal approximations iteratively.
• The algorithm achieves robust performance with low computational cost, making it competitive with traditional methods like Basis Pursuit and OMP.

CoSaMP: Iterative Signal Recovery from Incomplete and Inaccurate Samples

The paper "CoSaMP: Iterative Signal Recovery from Incomplete and Inaccurate Samples" by D. Needell and J.A. Tropp presents an innovative algorithm for addressing the challenge of signal recovery from compressive sampling, even when samples are noisy or incomplete. The proposed algorithm, CoSaMP (Compressive Sampling Matching Pursuit), is characterized by its iterative nature and the strong theoretical guarantees it offers regarding both performance and computational efficiency.

Overview of CoSaMP Algorithm

CoSaMP seeks to approximate an input signal that is known to be sparse or compressible. The algorithm operates iteratively, refining its approximation of the signal at each step. The major components of each iteration involve identification of significant signal components via a proxy, merging existing and newly identified supports, estimating the signal on the merged support through least-squares fitting, pruning to retain only the largest components, and updating the sample residual accordingly.

Key Theoretical Results

Reconstruction Guarantees

The authors establish rigorous bounds on the error for signal approximations produced by CoSaMP. Given a sampling matrix $\Phi$ satisfying the Restricted Isometry Property (RIP) with constant $\delta_{2s}$, and samples $u = \Phi x + e$ of an arbitrary signal $x$ corrupted with arbitrary noise $e$, CoSaMP produces a $2s$-sparse approximation $\hat{x}$ such that:

$$\|x - \hat{x}\|_2 \leq C \max \left\{ \eta, \frac{1}{\sqrt{s}} \| x - x_s \|_1 + \| e \|_2 \right\}$$

where $\eta$ is a precision parameter, $x_s$ is the best $s$-sparse approximation of $x$, and $C$ is a constant.

Computational Efficiency

CoSaMP is designed to be computationally efficient. The running time of the algorithm is dominated by matrix-vector multiplications, achieving a time complexity of $O(N \log^2 N)$ per iteration, where $N$ is the signal length. Additionally, the implementation requires only $O(N)$ storage, making it highly practical for large-scale problems.

Implications and Comparisons

CoSaMP offers several notable advantages compared with other signal recovery methods:

• General Applicability: The algorithm accepts a variety of sampling schemes, assumed to satisfy RIP, which encompasses both Gaussian random matrices and partial Fourier matrices.
• Optimal Sampling Efficiency: It requires a minimal number of samples, denoted by $m = O(s \log N)$, to recover $s$-sparse signals, placing it in the field of optimal compressive sensing algorithms.
• Uniform Error Bounds: The error bounds provided are uniform, meaning they hold for all signals and sampling matrices satisfying the given conditions.
• Robustness to Noise: The algorithm is robust in the presence of noise, gracefully degrading performance in proportion to the noise level.

Compared to traditional methods, such as Basis Pursuit via convex optimization, CoSaMP holds its own by providing similar error guarantees but with potentially lower computational costs. Greedy algorithms like Orthogonal Matching Pursuit (OMP) and Regularized Orthogonal Matching Pursuit (ROMP) do not consistently provide the uniform or optimal error guarantees that CoSaMP does. Sublinear time algorithms, while faster, typically have more restrictive sample requirements and achieve less accurate recoveries.

Future Directions

The capabilities of CoSaMP suggest several avenues for future research:

• Adaptive Parameter Schemes: Investigating adaptive methods for selecting algorithm parameters dynamically could enhance performance across varying conditions.
• Extended Applications: Applications beyond standard signal processing, such as in high-dimensional data analysis, can be explored to leverage CoSaMP's efficiency and robustness.
• Integration with Neural Networks: Incorporating CoSaMP within neural network architectures could potentially enhance model performance on tasks involving sparse data representations or signal recovery contexts.

Conclusion

CoSaMP presents a robust, efficient, and theoretically sound approach to signal recovery in the compressed sensing framework. Its balance of computational efficiency and strong recovery guarantees underlines its practical utility and establishes it as a valuable tool for both theoretical research and practical applications in signal processing and beyond.