Learning Topology-Driven Multi-Subspace Fusion for Grassmannian Deep Networks

Grassmannian manifold offers a powerful carrier for geometric representation learning by modelling high-dimensional data as low-dimensional subspaces. However, existing approaches predominantly rely on static single-subspace representations, neglecting the dynamic interplay between multiple subspaces critical for capturing complex geometric structures. To address this limitation, we propose a topologydriven multi-subspace fusion framework that enables adaptive subspace collaboration on the Grassmannian. Our solution introduces two key innovations: (1) Inspired by the Kolmogorov-Arnold representation theorem, an adaptive multi-subspace modelling mechanism is proposed that dynamically selects and weights task-relevant subspaces via topological convergence analysis, and (2) a multi-subspace interaction block that fuses heterogeneous geometric representations through Fréchet mean optimisation on the manifold. Theoretically, we establish the convergence guarantees of adaptive subspaces under a projection metric topology, ensuring stable gradient-based optimisation. Practically, we integrate Riemannian batch normalisation and mutual information regularisation to enhance discriminability and robustness. Extensive experiments on 3D action recognition (HDM05, FPHA), EEG classification (MAMEM-SSVEPII), and graph tasks demonstrate state-of-the-art performance. Our work not only advances geometric deep learning but also successfully adapts the proven multi-channel interaction philosophy of Euclidean networks to non-Euclidean domains, achieving superior discriminability and interpretability.