Differential Evolution for Grassmann Manifold Optimization: A Projection Approach
This provides a geometry-aware alternative for researchers in machine learning and signal processing dealing with subspace optimization, though it is incremental as it adapts an existing method to a specific manifold.
The authors tackled the problem of optimizing functions on Grassmann manifolds, which existing local methods struggle with due to nonconvex landscapes, by adapting Differential Evolution with a projection mechanism, resulting in a flexible algorithm that maintains feasibility and enables global exploration.
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.