GrassMLR: Rethinking Multinomial Logistic Regression on Grassmann Manifolds

Grassmann manifolds represent data as low-dimensional subspaces, effectively capturing the intrinsic geometric structure of high-dimensional data, thus achieving significant success in geometric representation learning. However, existing approaches typically map Riemannian features to Euclidean spaces for final decision, which inevitably introduces semantic distortions and separates the decision process from the underlying geometry, thereby overlooking the geometric power embedded in the Riemannian space. To address this issue, we propose **Grass**mann **M**ultinomial **L**ogistic **R**egression **(GrassMLR)**, which performs the classification decision directly on the Grassmann manifold. Specifically, to align classification decisions with the intrinsic geometry of the Grassmann manifold, GrassMLR introduces two learnable Grassmann anchors for each class. For an input subspace, it computes the squared geodesic distances to the two anchors. A distance-difference function is then defined as the difference between these two squared geodesic distances. This formulation induces a decision hypersurface with explicit geometric meaning, thereby ensuring that classification decisions remain consistent with the underlying manifold geometry. Furthermore, to obtain representations better suited to GrassMLR, we propose a Quotient-Space Representation Learning (QSRL) strategy. In particular, we exploit the quotient-space invariance of the Grassmann manifold to apply a learnable Lie-group transformation to subspace representations, preserving the quotient space while obtaining more effective geometric representations. To validate the generalised potential of our design, we apply GrassMLR to multiple downstream tasks. Extensive experiments demonstrate that GrassMLR and QSRL work synergistically to improve the performance of Grassmann neural networks and obtain more discriminative geometric representations. These results validate the effectiveness of geometry-aware decision making and representation optimisation under quotient space invariance.