Lp quasi-norm minimization: algorithm and applications

• Published in: EURASIP journal on advances in signal processing Vol. 2024; no. 1; pp. 22 - 28
• Main Authors: Sleem, Omar M., Ashour, M. E., Aybat, N. S., Lagoa, Constantino M.
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.12.2024; Springer; Springer Nature B.V; SpringerOpen
• Subjects: Algorithms; Alternating direction method of multipliers; Comparative analysis; Compressed sensing; Continuity (mathematics); Engineering; Heuristic methods; Iterative methods; Machine learning; Matrix completion; Operators (mathematics); Optimization; Polynomials; Quantum Information Technology; Rank minimization; Signal processing; Signal reconstruction; Signal,Image and Speech Processing; Sparsity; Spintronics; System identification; Alternating direction method of multipliers; Sparsity; Rank minimization; System identification; Compressed sensing; Matrix completion; Proximal gradient method
• ISSN: 1687-6180; 1687-6172; 1687-6180
• DOI: 10.1186/s13634-024-01114-6

Sparsity finds applications in diverse areas such as statistics, machine learning, and signal processing. Computations over sparse structures are less complex compared to their dense counterparts and need less storage. This paper proposes a heuristic method for retrieving sparse approximate solutions of optimization problems via minimizing the ℓ p quasi-norm, where 0 < p < 1 . An iterative two-block algorithm for minimizing the ℓ p quasi-norm subject to convex constraints is proposed. The proposed algorithm requires solving for the roots of a scalar degree polynomial as opposed to applying a soft thresholding operator in the case of ℓ 1 norm minimization. The algorithm’s merit relies on its ability to solve the ℓ p quasi-norm minimization subject to any convex constraints set. For the specific case of constraints defined by differentiable functions with Lipschitz continuous gradient, a second, faster algorithm is proposed. Using a proximal gradient step, we mitigate the convex projection step and hence enhance the algorithm’s speed while proving its convergence. We present various applications where the proposed algorithm excels, namely, sparse signal reconstruction, system identification, and matrix completion. The results demonstrate the significant gains obtained by the proposed algorithm compared to other ℓ p quasi-norm based methods presented in previous literature.