A Riemannian quasi-Newton algorithm for optimization with Euclidean bounds

2026-05-11

AI Summary

This work addresses Riemannian optimization problems with Euclidean bound constraints—common in covariance estimation, neuroimaging, and signal processing—and proposes the first method extending the generalized Cauchy point strategy from L-BFGS-B to Riemannian manifolds, enabling joint optimization over bounded Euclidean variables and manifold variables. The approach integrates limited-memory quasi-Newton updates in the tangent space with a boundary-handling mechanism and is implemented within the Manopt.jl framework. Experimental results demonstrate that the method matches the performance of classical L-BFGS-B, substantially outperforms interior-point methods, and achieves speedups of several orders of magnitude in blind source separation and principal component analysis tasks.

Abstract

We propose a Riemannian limited-memory BFGS method for optimization problems with Euclidean bounds. The method combines a limited-memory quasi-Newton update in the tangent space with a Riemannian adaptation of the generalized Cauchy point strategy from classical L-BFGS-B, enabling efficient handling of Euclidean bounds while exploiting the geometric structure of the optimization domain. This setting is important in several applications, including covariance matrix estimation with bounded variance, neuroimaging, EEG signal classification, and other signal processing or computer-vision tasks that couple manifold variables with constrained Euclidean parameters. We provide a generic algorithmic framework and an implementation of the algorithm in the Manopt.jl library. Numerical experiments on benchmark problems indicate only minor reduction in performance on Euclidean problems compared to the classical L-BFGS-B method, while outperforming interior-point methods. Furthermore, the algorithm was tested on two mixed manifold and bounded Euclidean problems: amplitude-limited blind source separation with Gaussianity penalization and bounded-variance maximum likelihood common principal components analysis. The proposed method outperforms existing methods by several orders of magnitude.

Problem

• Riemannian optimization
• Euclidean bounds
• constrained optimization
• manifold optimization
• bounded variables

Innovation

• Riemannian optimization
• L-BFGS-B
• bounded constraints
• quasi-Newton method
• Manopt.jl