Lp quasi-norm minimization: algorithm and applications

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 solution...

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Veröffentlicht in:	EURASIP journal on advances in signal processing Jg. 2024; H. 1; S. 22 - 28
Hauptverfasser:	Sleem, Omar M., Ashour, M. E., Aybat, N. S., Lagoa, Constantino M.
Format:	Journal Article
Sprache:	Englisch
Veröffentlicht:	Cham Springer International Publishing 01.12.2024 Springer Springer Nature B.V SpringerOpen
Schlagworte:	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
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Abstract	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.
AbstractList	Abstract 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 $$\ell _{p}$$ ℓ p quasi-norm, where $$0<p<1$$ 0 < p < 1 . An iterative two-block algorithm for minimizing the $$\ell _{p}$$ 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 $$\ell _{1}$$ ℓ 1 norm minimization. The algorithm’s merit relies on its ability to solve the $$\ell _{p}$$ 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 $$\ell _{p}$$ ℓ p quasi-norm based methods presented in previous literature. 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 ℓp quasi-norm, where 0<p<1. An iterative two-block algorithm for minimizing the ℓp 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 ℓ1 norm minimization. The algorithm’s merit relies on its ability to solve the ℓp 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 ℓp quasi-norm based methods presented in previous literature. 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 [Formula omitted] quasi-norm, where [Formula omitted]. An iterative two-block algorithm for minimizing the [Formula omitted] 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 [Formula omitted] norm minimization. The algorithm's merit relies on its ability to solve the [Formula omitted] 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 [Formula omitted] quasi-norm based methods presented in previous literature. 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 $$\ell _{p}$$ ℓ p quasi-norm, where $$0<p<1$$ 0 < p < 1 . An iterative two-block algorithm for minimizing the $$\ell _{p}$$ 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 $$\ell _{1}$$ ℓ 1 norm minimization. The algorithm’s merit relies on its ability to solve the $$\ell _{p}$$ 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 $$\ell _{p}$$ ℓ p quasi-norm based methods presented in previous literature. 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.
ArticleNumber	22
Audience	Academic
Author	Ashour, M. E. Lagoa, Constantino M. Sleem, Omar M. Aybat, N. S.
Author_xml	– sequence: 1 givenname: Omar M. orcidid: 0000-0002-4994-0808 surname: Sleem fullname: Sleem, Omar M. email: oms46@psu.edu organization: Department of Electrical Engineering, Pennsylvania State University – sequence: 2 givenname: M. E. surname: Ashour fullname: Ashour, M. E. organization: Wireless R&D Department, Qualcomm Technologies, Inc – sequence: 3 givenname: N. S. surname: Aybat fullname: Aybat, N. S. organization: Department of Industrial and Manufacturing Engineering, Pennsylvania State University – sequence: 4 givenname: Constantino M. surname: Lagoa fullname: Lagoa, Constantino M. organization: Department of Electrical Engineering, Pennsylvania State University
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Snippet	Sparsity finds applications in diverse areas such as statistics, machine learning, and signal processing. Computations over sparse structures are less complex... Abstract Sparsity finds applications in diverse areas such as statistics, machine learning, and signal processing. Computations over sparse structures are less...
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SubjectTerms	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
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Title	Lp quasi-norm minimization: algorithm and applications
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