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

TL;DR

This work addresses the limitation of single-subspace representations on Grassmannian manifolds by introducing a topology-driven adaptive multi-subspace fusion framework. It combines an Adaptive Multi-Subspace Encoder (AMSE) with a Multi-Subspace Interaction Block to dynamically construct and fuse multiple subspaces while preserving manifold geometry, supported by convergence guarantees under a metric topology and Fréchet-mean based fusion. The approach achieves state-of-the-art results across 3D action recognition, EEG classification, and graph tasks, while offering improved robustness and interpretability through Riemannian batch normalization and a mutual-information inspired regularizer. Overall, the topology-guided multi-subspace fusion advances geometric deep learning on non-Euclidean domains and demonstrates strong cross-domain applicability.

Abstract

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 topology-driven 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.

Figures (7)

• Figure 1: Illustration of the problem evolution. (a) shows semantic distortion of features in Euclidean space. (b) reveals semantic constraint using a single Grassmannian subspace. (c) presents the proposed framework incorporating multi-Grassmannian subspace fusion.
• Figure 2: Overview of GMSF-Net. (a) Architecture of GMSF-Net. (b) Structure of the Adaptive Multi-Subspace Encoder (AMSE). (c) Fine-Grained Design. The proposed efficient Adaptive Multi-Subspace Construction (AdaMSC). (d) Subspace Interaction Design. The proposed discriminative Multi-Subspace Interaction Block.
• Figure 3: Illustration of one iteration of the fusion between multiple subspaces. The data points $P_i$, each representing a different subspace, are logarithmically mapped to the tangent space at $\Gamma$. These subspaces $S_i$ are then arithmetically averaged. The result is exponentially mapped back to the manifold, yielding the updated mean $\Gamma_{\text{new}}$.
• Figure 4: Based on GMSF-Net, the heatmaps of absolute gradient responses (computed following the method in pan2022matt) are presented across five frequency categories on the MAMEM-SSVEP-II dataset (S11). In each heatmap, the x-axis represents time, and the y-axis corresponds to EEG channels.
• Figure 5: The spatial topomaps of the mean absolute gradient responses across time for the S11 subject on the MAMEM-SSVEP-II dataset, based on GMSF-Net (computed as in pan2022matt. The brain region marked in dark red corresponds to channel Oz, which exhibits strong gradient activation across the visual cortex under all stimulation frequencies.

Theorems & Definitions (17)

• Definition 1: Riemannian Manifold
• Definition 2: Pullback Metrics
• Definition 3: Tangent Space in the Orthonormal Basis View
• Definition 4: Exponential Map in the Orthogonal Basis Viewpoint
• Definition 5: Logarithmic Map in the Orthogonal Basis Viewpoint
• Definition 6: Topology Space Definition
• Definition 7: Metric Topology Definition
• Theorem 1
• proof
• Proposition 2