Learning on Adaptive Manifolds for Graph Collaborative Filtering

Graph-based collaborative filtering has advanced by modeling higher-order interactions, yet performance remains constrained by underlying geometric assumptions and propagation schemes. User-item interaction graphs typically exhibit pronounced topological heterogeneity, whereas existing methods rely on a fixed, homogeneous geometry and employ tangent space aggregation. To address these fundamental limitations, this paper introduces Adaptive Geometric Collaborative Filtering (AGCF), a novel method rooted in Hamiltonian dynamics, which reframes representation learning as a physical process evolving on a time-varying manifold. AGCF is distinguished by an integrated design comprising: (1) a learnable, node-dependent Riemannian metric that construct a continuous heterogeneous manifold aligned with local topology; (2) unified dynamic trajectories that achieve intrinsic propagation without tangent space approximations; (3) a channel-wise metric that captures semantic anisotropy in the feature space. We rigorously prove global existence and uniqueness of the induced dynamics and explain the mechanism enabling long-range information propagation. Extensive experiments on five benchmark datasets show consistent gains over representative baselines.