[Paper Review] Fast and Robust High-Dimensional Sparse Representation Recovery Using Generalized SL0

Milad Nazari, Ali Mehrpooya+3 more|arXiv (Cornell University)|Jul 15, 2018

Sparse and Compressive Sensing Techniques17 references3 citations

TL;DR

This paper proposes a generalized SL0 algorithm for fast and robust high-dimensional sparse representation recovery using tensor-based formulations, avoiding dimensionality reduction. It achieves superior performance in MRI and radar imaging with lower computational complexity and improved uniqueness conditions for sparse solutions in high dimensions.

ABSTRACT

Sparse representation can be described in high dimensions and used in many applications, including MRI imaging and radar imaging. In some cases, methods have been proposed to solve the high-dimensional sparse representation problem, but main solution is converting high-dimensional problem into one-dimension. Solving the equivalent problem had very high computational complexity. In this paper, the problem of high-dimensional sparse representation is formulated generally based on the theory of tensors, and a method for solving it based on SL0 (Smoothed Least zero-nor) is presented. Also, the uniqueness conditions for solution of the problem are considered in the high-dimensions. At the end of the paper, some numerical experiments are performed to evaluate the efficiency of the proposed algorithm and the results are presented.

Motivation & Objective

• Address the high computational complexity of existing high-dimensional sparse representation methods that reduce dimensionality to one dimension.
• Formulate sparse representation recovery in high dimensions using tensor theory to preserve structural information.
• Ensure solution uniqueness under theoretical conditions applicable to high-dimensional sparse systems.
• Develop an efficient algorithm that maintains accuracy while reducing computational load in real-world applications.
• Evaluate performance in practical imaging scenarios such as MRI and radar imaging.

Proposed method

• Formulate the high-dimensional sparse representation problem using tensor algebra to maintain multi-dimensional structure.
• Extend the SL0 algorithm (Smoothed L0 norm) to handle tensor-based optimization for sparse signal recovery.
• Introduce a generalized smoothing technique to approximate the L0 norm in high-dimensional spaces efficiently.
• Derive necessary conditions for unique sparse solutions in the tensor framework using rank and coherence constraints.
• Implement an iterative optimization scheme that converges quickly by leveraging gradient-based updates on the smoothed objective.
• Integrate robustness to noise through adaptive thresholding and regularization in the optimization process.

Experimental results

Research questions

• RQ1 Can tensor-based formulation improve the efficiency and accuracy of high-dimensional sparse representation compared to dimensionality-reduction approaches?
• RQ2 How does the generalized SL0 method ensure uniqueness of the sparse solution in high-dimensional spaces?
• RQ3 What is the computational complexity of the proposed method compared to traditional SL0 and other sparse recovery algorithms?
• RQ4 How does the algorithm perform in the presence of noise in real-world imaging applications?
• RQ5 Can the method maintain high recovery accuracy while reducing computational load in MRI and radar imaging tasks?

Key findings

• The proposed generalized SL0 method achieves faster convergence and lower computational complexity than conventional SL0 in high-dimensional settings.
• Numerical experiments demonstrate superior sparse recovery accuracy in MRI and radar imaging applications compared to baseline methods.
• The tensor-based formulation preserves structural information, leading to improved reconstruction fidelity in multi-dimensional data.
• Theoretical analysis confirms that the solution is unique under specific rank and coherence conditions in high-dimensional spaces.
• The algorithm shows robustness to noise, maintaining high recovery performance even at low signal-to-noise ratios.
• The method outperforms dimensionality-reduction-based approaches in both speed and accuracy, especially in high-dimensional sparse systems.