Riemannian optimization using three different metrics for Hermitian PSD fixed-rank constraints

Shixin Zheng (), Wen Huang (), Bart Vandereycken () and Xiangxiong Zhang ()
Additional contact information
Shixin Zheng: Purdue University
Wen Huang: Xiamen University
Bart Vandereycken: University of Geneva
Xiangxiong Zhang: Purdue University

Computational Optimization and Applications, 2025, vol. 91, issue 3, No 5, 1135-1184

Abstract: Abstract For smooth optimization problems with a Hermitian positive semidefinite fixed-rank constraint, we consider three existing approaches including the simple Burer–Monteiro method, and Riemannian optimization over quotient geometry and the embedded geometry. These three methods can be all represented via quotient geometry with three Riemannian metrics $$g^i(\cdot , \cdot )$$ g i ( · , · ) $$(i=1,2,3)$$ ( i = 1 , 2 , 3 ) . By taking the nonlinear conjugate gradient method (CG) as an example, we show that CG in the factor-based Burer–Monteiro approach is equivalent to Riemannian CG on the quotient geometry with the Bures–Wasserstein metric $$g^1$$ g 1 . Riemannian CG on the quotient geometry with the metric $$g^3$$ g 3 is equivalent to Riemannian CG on the embedded geometry. For comparing the three approaches, we analyze the condition number of the Riemannian Hessian near a minimizer under the three different metrics. Under certain assumptions, the condition number from the Bures–Wasserstein metric $$g^1$$ g 1 is significantly worse than the other two metrics. Numerical experiments show that the Burer–Monteiro CG method has obviously slower asymptotic convergence rate either when the minimizer has a large condition number or when it is rank deficient, which is consistent with the condition number analysis.

Keywords: Riemannian optimization; Hermitian PSD fixed-rank matrices; Embedded manifold; Quotient manifold; Burer–Monteiro; Conjugate gradient; Riemannian Hessian; Bures–Wasserstein metric (search for similar items in EconPapers)
Date: 2025
References: View complete reference list from CitEc
Citations:

Downloads: (external link)
http://link.springer.com/10.1007/s10589-025-00687-8 Abstract (text/html)
Access to the full text of the articles in this series is restricted.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:coopap:v:91:y:2025:i:3:d:10.1007_s10589-025-00687-8

Ordering information: This journal article can be ordered from
http://www.springer.com/math/journal/10589

DOI: 10.1007/s10589-025-00687-8

Access Statistics for this article

Computational Optimization and Applications is currently edited by William W. Hager

More articles in Computational Optimization and Applications from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().