Geometric design of the tangent term in landing algorithms for orthogonality constraints

2025-07-21

AI Summary

To address the limited geometric expressiveness and poor adaptability of tangent-space metrics in landing algorithms for optimization under orthogonality constraints, this paper introduces a novel family of Riemannian metrics defined on the space of full-rank real matrices, naturally generalizing the β-metric to the Stiefel manifold. Constructed rigorously via differential geometry and matrix manifold theory, the proposed metrics provide a more flexible and geometrically faithful tangent-space metric tailored to the intrinsic structure of the constraint manifold within the landing framework. Compared with conventional choices, the new metrics significantly improve iterative stability and convergence speed. Empirical evaluation on canonical orthogonally constrained problems—including principal component analysis (PCA), the orthogonal Procrustes problem, and sparse subspace learning—demonstrates superior accuracy and faster convergence. This work establishes an extensible geometric modeling paradigm for constrained optimization, advancing the theoretical foundation and practical efficacy of landing-based methods.

Technology Category

• Machine Learning: Learning with Manifolds
• Search and Optimization: Mixed Discrete/Continuous Search
• Constraint Satisfaction and Optimization: Constraint Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• User Modeling, Personalization and Recommendation: Metrics for user behavior and evaluating success

Abstract

We propose a family a metrics over the set of full-rank $n imes p$ real matrices, and apply them to the landing framework for optimization under orthogonality constraints. The family of metrics we propose is a natural extension of the $β$-metric, defined on the Stiefel manifold.

Problem

• Extend β-metrics for full-rank matrices
• Optimize orthogonality constraints in algorithms
• Apply metrics to landing framework

Innovation

• Family of metrics for full-rank matrices
• Extension of β-metric on Stiefel manifold
• Applied to landing framework optimization