A fast approach for overcomplete sparse decomposition based on smoothed L0 norm (0809.2508v2)

Abstract: In this paper, a fast algorithm for overcomplete sparse decomposition, called SL0, is proposed. The algorithm is essentially a method for obtaining sparse solutions of underdetermined systems of linear equations, and its applications include underdetermined Sparse Component Analysis (SCA), atomic decomposition on overcomplete dictionaries, compressed sensing, and decoding real field codes. Contrary to previous methods, which usually solve this problem by minimizing the L1 norm using Linear Programming (LP) techniques, our algorithm tries to directly minimize the L0 norm. It is experimentally shown that the proposed algorithm is about two to three orders of magnitude faster than the state-of-the-art interior-point LP solvers, while providing the same (or better) accuracy.

• The paper presents the SL0 algorithm, which directly minimizes a smoothed L0 norm to reduce the computational complexity of sparse recovery.
• It leverages a steepest ascent with a projection step, starting from the L2 norm solution and iteratively enhancing sparsity.
• Extensive simulations reveal that SL0 achieves up to three orders of magnitude speed improvement and robust noise resistance compared to LP techniques.

A Fast Approach for Overcomplete Sparse Decomposition Based on Smoothed L0 Norm

The paper "A fast approach for overcomplete sparse decomposition based on smoothed $L0$ norm" by Mohimani, Babaie-Zadeh, and Jutten, proposes an efficient algorithm, termed as SL0, for obtaining sparse solutions of underdetermined systems of linear equations. This method holds significant advantages over previous algorithms primarily based on Linear Programming (LP) techniques.

Core Contribution

The primary contribution of this paper revolves around a novel method named SL0, designed for sparse decomposition in overcomplete dictionaries. The method directly targets the minimization of the $L0$ norm, in contrast to conventional LP approaches that approximate $L0$ minimization with $L1$ minimization. The algorithm's working principle involves approximating the discontinuous $L0$ norm with a continuous function, subsequently applying a minimization algorithm for continuous functions. This notably mitigates the computational complexity typically associated with combinatorial searches in $L0$ minimization.

Algorithm Details

SL0 functions by initializing the sparse solution as the minimum $L2$ norm solution. The notion hinges on the fact that for significantly large values of the smoothing parameter $\sigma$, the $L0$ minimization problem tends to the $L2$ norm solution. Thereafter, a sequence of gradually decreasing $\sigma$ guides the algorithm towards a sparser solution.

A fundamental aspect of the SL0 algorithm is its adoption of a steepest ascent method combined with a projection step, iteratively improving the sparsity of the solution while maintaining the feasibility of the linear constraints. The theoretical underpinnings ensure that for a small enough $\sigma$, the algorithm's output converges to the sparsest solution.

Numerical Performance and Robustness

In extensive simulations, the SL0 algorithm exhibited performance enhancements of two to three orders of magnitude in speed compared to state-of-the-art LP solvers, without compromising accuracy. For instance, while solving a decomposition problem for $m=1000$, $n=400$, and $k=100$ (with $p=0.1$ sparsity factor), SL0 consistently achieved higher SNR values faster than both LP-based methods and FOCUSS.

Additionally, the algorithm displayed robustness to noise, demonstrating superior performance over LP methods, especially under moderate noise conditions. This robustness is credited to the algorithm's inherent handling of the smoothed norm, which is less sensitive to minute variations than the $L0$ norm.

Key Insights and Implications

Several salient points emerge from this research, impacting both practical applications and theoretical advancements:

1. Scalability: SL0 is highly scalable, proving effective even as the dimensions of the problem (number of sensors and sources) increase. This makes it particularly suitable for high-dimensional signal processing tasks.
2. Sparsity Handling: The algorithm excels in scenarios where the number of non-zero components is near the theoretical limit ($n/2$), achieving performant results closer to the sparsest solutions.
3. Computational Efficiency: With the ability to substantially reduce computational overhead, SL0 is a practical alternative to LP solvers for real-time applications requiring real-time sparse recovery.

Theoretical Developments

From a theoretical perspective, three pivotal theorems are introduced:

• Initialization with $L2$: Theorem 2 justifies the use of the minimum $L2$-norm solution for initializing the algorithm. It demonstrates how for sufficiently large $\sigma$, SL0's initial solution naturally converges to the $L2$-norm minimizer.
• Convergence to Sparsest Solution: Theorem 1 establishes that the sequence of solutions obtained by gradually decreasing $\sigma$ converges to the sparsest solution, enabling the effective minimization of the $L0$ norm.
• Robustness to Noise: Theorem 3 extends this to noisy scenarios, providing bounds on the estimation error in relation to noise power, and demonstrating SL0's potential for robust performance under noisy conditions.

Future Directions

The promising results and theoretical assurances presented in this paper pave the way for several future research directions:

• Enhanced Noise Handling: Exploring strategies to incorporate noise considerations directly within the algorithm could further enhance SL0's robustness.
• Optimal Parameter Selection: Developing adaptive methods to choose optimal parameters like the sequence of $\sigma$ values could automate and potentially improve the convergence speed and accuracy of the algorithm.
• Application to Real-World Data: Testing and fine-tuning SL0 for specific real-world applications, such as in compressed sensing and image processing, could solidify its utility and effectiveness in practical scenarios.

In summary, this paper contributes significantly to the domain of sparse signal recovery, presenting a theoretically sound and practically efficient algorithm that supersedes existing methods in both speed and robustness.