A Riemannian Optimization Approach to Clustering Problems

This paper considers the optimization problem min X ∈ F v f ( X ) + λ ‖ X ‖ 1 , where f is smooth, F v = { X ∈ R n × q : X T X = I q , v ∈ span ( X ) } , and v is a given positive vector. The clustering models including but not limited to the models used by k -means, community detection, and normali...

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Bibliographic Details
Published in:Journal of scientific computing Vol. 103; no. 1; p. 8
Main Authors: Huang, Wen, Wei, Meng, Gallivan, Kyle A., Van Dooren, Paul
Format: Journal Article
Language:English
Published: New York Springer US 01.04.2025
Springer Nature B.V
Subjects:
Algorithms
Clustering
Computational Mathematics and Numerical Analysis
Image segmentation
Linear algebra
Manifolds (mathematics)
Mathematical and Computational Engineering
Mathematical and Computational Physics
Mathematics
Mathematics and Statistics
Modularity
Optimization
Signal processing
Sparsity
Theoretical
Inexact proximal gradient method
Manifold recognition
Clustering
Riemannian optimization
ISSN:0885-7474, 1573-7691
Online Access:Get full text
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Summary:This paper considers the optimization problem min X ∈ F v f ( X ) + λ ‖ X ‖ 1 , where f is smooth, F v = { X ∈ R n × q : X T X = I q , v ∈ span ( X ) } , and v is a given positive vector. The clustering models including but not limited to the models used by k -means, community detection, and normalized cut can be reformulated as such optimization problems. It is proven that the domain F v forms a compact embedded submanifold of R n × q and optimization-related tools including a family of computationally efficient retractions and an orthonormal basis of any normal space of F v are derived. A Riemannian proximal gradient method that allows an adaptive step size is proposed. The proposed Riemannian proximal gradient method solves its subproblem inexactly and still guarantees its global convergence. Numerical experiments on community detection in networks and normalized cut for image segmentation are used to demonstrate the performance of the proposed method.
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ISSN:0885-7474
1573-7691
DOI:10.1007/s10915-025-02806-3