Hamiltonian-Structured Interaction Fields for Joint Cross-Mode Representation Dynamics

Tensor factorization provides a compact framework for modeling high-order data by learning low-dimensional mode-wise representations and composing them for individual tensor elements. Neural tensor factorization further enriches this paradigm through nonlinear cross-mode interaction, yet such flexible fusion does not explicitly organize how changes in one mode influence the refinement of others. To address this limitation, we propose the Hamiltonian-Structured Neural Tensor Factorization (HSNTF), which introduces a Hamiltonian-structured interaction field for context-specific latent representation refinement. For each tensor element, HSNTF assembles the corresponding mode-wise embeddings into a joint latent representation. A scalar generator induces a Hamiltonian vector field that governs the evolution of the joint representation. Its Jacobian maps second-order curvature, including mixed derivatives, into explicit local cross-mode influence operators. The behavior of these operators is structurally coupled during representation refinement. Short-horizon multi-step evolution repeatedly applies the field to the updated latent state to produce context-adapted representations. Experiments on five real-world tensor datasets demonstrate consistent predictive gains. Structural ablations, perturbation analyses, and multi-step interventions show that the learned field captures local cross-mode responses and enables mediated cross-mode information transfer through multi-step integration. Controlled experiments further show that shared directional structure is particularly useful under scarce supervision, with its benefit depending on structural compatibility, sample size, and noise. These results establish Hamiltonian structure as a unified mechanism for organizing coupled cross-mode representation dynamics, without assuming physical dynamics in the observed tensor data.