Differential Evolution for Grassmann Manifold Optimization: A Projection Approach

TL;DR

This work addresses the challenge of global optimization for functions defined on the Grassmann manifold $Gr(k,n)$, where traditional Riemannian methods are inherently local. It proposes a geometry-aware, global search via Differential Evolution (DE) adapted to $Gr(k,n)$ with a QR-based projection to maintain feasibility and self-adaptive parameter control to balance exploration and exploitation. The method is validated on five Grassmannian objectives, including PCA, subspace alignment, and subspace clustering, under moderate dimension ($n=20$, $k=5$, $d=100$), demonstrating accurate recovery of subspaces and competitive global search performance. The results highlight a robust, scalable alternative to local optimization for subspace estimation, dimensionality reduction, and low-rank modeling in ML and signal processing.

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.