Generalized Orthogonal Matching Pursuit (1111.6664v2)

Abstract: As a greedy algorithm to recover sparse signals from compressed measurements, orthogonal matching pursuit (OMP) algorithm has received much attention in recent years. In this paper, we introduce an extension of the OMP for pursuing efficiency in reconstructing sparse signals. Our approach, henceforth referred to as generalized OMP (gOMP), is literally a generalization of the OMP in the sense that multiple $N$ indices are identified per iteration. Owing to the selection of multiple ''correct'' indices, the gOMP algorithm is finished with much smaller number of iterations when compared to the OMP. We show that the gOMP can perfectly reconstruct any $K$-sparse signals ($K > 1$), provided that the sensing matrix satisfies the RIP with $\delta_{NK} < \frac{\sqrt{N}}{\sqrt{K} + 3 \sqrt{N}}$. We also demonstrate by empirical simulations that the gOMP has excellent recovery performance comparable to $\ell_1$-minimization technique with fast processing speed and competitive computational complexity.

• The paper introduces a multi-index extension to the classic OMP algorithm for enhanced sparse signal recovery.
• It establishes theoretical recovery guarantees via RIP conditions and shows competitive empirical performance compared to other greedy methods.
• The method reduces iterations and computational complexity, making it suitable for real-time, high-dimensional signal processing applications.

Generalized Orthogonal Matching Pursuit

The paper "Generalized Orthogonal Matching Pursuit" addresses the enhancement of the classic Orthogonal Matching Pursuit (OMP) algorithm for efficient sparse signal recovery from compressed measurements. Sparse signal recovery is a fundamental aspect of compressive sensing (CS), where the objective is to reconstruct sparse signals from fewer measurements than traditionally required by Nyquist sampling theory.

The authors introduce an extension to the OMP algorithm, referred to as Generalized Orthogonal Matching Pursuit (gOMP). The main modification involves selecting multiple indices in each iteration instead of a single one, which the traditional OMP implements. This generalization potentially allows the gOMP algorithm to identify multiple correct indices per iteration, resulting in a reduced number of iterations necessary for signal recovery.

Key Contributions

1. Algorithm Design: The gOMP algorithm selects $N$ indices at each iteration, where $N \geq 1$, as opposed to the single index selection in OMP. This accelerates the convergence of the algorithm and enhances computational efficiency.
2. Recovery Guarantee: The paper derives a theoretical condition under which gOMP can perfectly recover any $K$-sparse signal. Specifically, it shows that if the sensing matrix satisfies the Restricted Isometry Property (RIP) with a constant $$\delta_{NK} < \frac{\sqrt{N}}{\sqrt{K} + 3\sqrt{N}}$$, then exact recovery is achievable.
3. Empirical Performance: Through simulations, it is demonstrated that gOMP offers recovery performance comparable to $\ell_1$-minimization techniques and outperforms other greedy approaches such as CoSaMP and StOMP when $N$ is optimally chosen. The numerical tests show substantial reductions in computational complexity and running time compared to OMP, especially for signals with a moderately large sparsity level.
4. Complexity Analysis: The authors provide a comprehensive analysis of the computational complexity of the gOMP algorithm, which is expressed as approximately $2s mn + (2N^2 + N)s^2 m$ flops, where $s$ is the number of iterations and $m$, $n$ are the dimensions of the sensing matrix.

Implications and Future Directions

The introduction of the gOMP algorithm has substantial practical implications, especially in real-time applications requiring fast processing. The parallel nature of the multiple index selection lends itself to implementations on modern computing architectures with multiple processors or cores. Additionally, the ability to reduce iteration count directly translates to faster execution times, making the algorithm appealing for use cases involving data with high dimensionality.

From a theoretical perspective, the recovery conditions and complexity bounds offer valuable insights into the behavior of greedy algorithms under the framework of compressive sensing. While the derived RIP condition is conservative, indicating the worst-case scenario, the practical performance is often superior due to multiple correct indices being found in fewer iterations.

Conclusion

The gOMP algorithm represents a significant advancement in the practical implementation of sparse signal recovery techniques. Its ability to balance accuracy and efficiency makes it a suitable candidate for various applications within signal processing and beyond, where rapid and precise recovery of sparse signals is a priority. The results in the paper open the door for further research into optimizing the number of indices selected per iteration and exploring the integration of gOMP within broader signal processing frameworks. Future research could focus on probabilistic recovery analyses or adaptive schemes to dynamically choose $N$, based on signal characteristics and processing constraints.