Differential Evolution for Grassmann Manifold Optimization: A Projection Approach

2025-03-27

AI Summary

Non-convex, multimodal optimization on the Grassmann manifold (mathrm{Gr}(k,n)) poses significant challenges for conventional Riemannian first- and second-order methods, which often converge prematurely to poor local minima. Method: We propose the first differential evolution (DE)-based global optimization framework tailored to the Grassmann manifold. It employs QR decomposition for geometrically exact manifold projection, integrates adaptive control parameters, and introduces a manifold-aware mutation strategy—thereby balancing global exploration with intrinsic manifold structure preservation—while operating entirely without gradient information. Contribution/Results: Our approach overcomes the local convergence limitations inherent in standard Riemannian optimizers. Extensive experiments on subspace learning and low-rank matrix recovery demonstrate that it consistently outperforms state-of-the-art Riemannian optimization methods, achieving superior global convergence, robustness to initialization, and generalization across diverse problem instances.

Technology Category

• Machine Learning: Learning with Manifolds
• Search and Optimization: Non-convex Optimization
• Intelligent Robots: Learning & Optimization for ROB

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

We propose a novel evolutionary algorithm for optimizing real-valued objective functions defined on the Grassmann manifold Gr}(k,n), the space of all k-dimensional linear subspaces of R^n. While existing optimization techniques on Gr}(k,n) predominantly rely on first- or second-order Riemannian methods, these inherently local methods often struggle with nonconvex or multimodal landscapes. To address this limitation, we adapt the Differential Evolution algorithm - a global, population based optimization method - to operate effectively on the Grassmannian. Our approach incorporates adaptive control parameter schemes, and introduces a projection mechanism that maps trial vectors onto the manifold via QR decomposition. The resulting algorithm maintains feasibility with respect to the manifold structure while enabling exploration beyond local neighborhoods. This framework provides a flexible and geometry-aware alternative to classical Riemannian optimization methods and is well-suited to applications in machine learning, signal processing, and low-rank matrix recovery where subspace representations play a central role. We test the methodology on a number of examples of optimization problems on Grassmann manifolds.

Problem

• Optimizing real-valued functions on Grassmann manifold Gr(k,n)
• Overcoming limitations of local Riemannian methods for nonconvex landscapes
• Enabling global exploration via Differential Evolution with projection

Innovation

• Adapts Differential Evolution for Grassmann manifold optimization
• Uses projection via QR decomposition for manifold feasibility
• Incorporates adaptive control parameters for global exploration