A Quantum-Behaved Particle Swarm Optimization Algorithm on Riemannian Manifolds

• Published in: Mathematics (Basel) Vol. 10; no. 22; p. 4168
• Main Authors: Halimu, Yeerjiang, Zhou, Chao, You, Qi, Sun, Jun
• Format: Journal Article
• Language: English
• Published: Basel MDPI AG 01.11.2022
• Subjects: Adaptation; Algorithms; Approximation; Computer vision; Conjugate gradient method; Data mining; Euclidean geometry; Euclidean space; Lie groups; Machine learning; Manifolds (Mathematics); Mathematical analysis; Mathematical functions; Mathematical optimization; matrix manifold optimization; Methods; Newton methods; Operators (mathematics); Optimization algorithms; Particle swarm optimization; Physics; Quantum theory; quantum-behaved particle swarm optimization; retracement operator; Riemann manifold; Riemannian manifold; Search process; Semidefinite programming; Steepest descent method; Swarm intelligence; Vectors (mathematics); China
• ISSN: 2227-7390; 2227-7390
• DOI: 10.3390/math10224168

The Riemannian manifold optimization algorithms have been widely used in machine learning, computer vision, data mining, and other technical fields. Most of these algorithms are based on the geodesic or the retracement operator and use the classical methods (i.e., the steepest descent method, the conjugate gradient method, the Newton method, etc.) to solve engineering optimization problems. However, they lack the ability to solve non-differentiable mathematical models and ensure global convergence for non-convex manifolds. Considering this issue, this paper proposes a quantum-behaved particle swarm optimization (QPSO) algorithm on Riemannian manifolds named RQPSO. In this algorithm, the quantum-behaved particles are randomly distributed on the manifold surface and iteratively updated during the whole search process. Then, the vector transfer operator is used to translate the guiding vectors, which are not in the same Euclidean space, to the tangent space of the particles. Through the searching of these guiding vectors, we can achieve the retracement and update of points and finally obtain the optimized result. The proposed RQPSO algorithm does not depend on the expression form of a problem and could deal with various engineering technical problems, including both differentiable and non-differentiable ones. To verify the performance of RQPSO experimentally, we compare it with some traditional algorithms on three common matrix manifold optimization problems. The experimental results show that RQPSO has better performance than its competitors in terms of calculation speed and optimization efficiency.