Learning Distributions of Grassmann Subspace Combinations for Effective Geometric Representations

Grassmann manifolds provide a principled framework for modelling data as low-dimensional subspaces, capturing intrinsic geometric structure and achieving considerable success in geometric representation learning. Existing approaches, however, typically treat all inner subspace combinations within a Grassmann point as equally important, overlooking their non-uniform distribution. To address this issue, we theoretically characterise the distribution of subspace combinations within a Grassmann point based on spectral measure theory and quadratic utility theory, revealing that different combinations contribute non-uniformly to representation. This insight suggests that Grassmann representation learning should adaptively model such distributions and exploit compatibility among low-dimensional subspaces to select and construct effective combinations. To this end, we propose a **Gra**ph-**S**tructured adaptive **S**ubSpace (GraSS) modelling on Grassmann manifolds. GraSS builds a relational graph among constituent subspaces to jointly capture their importance and structural compatibility. Based on the constructed graph, we identify the maximum connected subgraph with high coherence and accordingly adaptively determine the mutually compatible constituent subspaces and their composition. Furthermore, we introduce an adaptive gating selection mechanism to dynamically filter the less powerful combinations, suppressing redundant combinations while emphasising those with higher representation utility. Finally, we apply GraSS to multiple downstream tasks. Extensive experiments show that GraSS consistently improves existing Grassmann representation learning methods across tasks and exhibits strong generalisation.