HC-SPA: Hyperbolic Cosine-Based Symplectic Phase Alignment for Fusion Optimization.

In multimodal collaborative learning, the gradient dynamics of heterogeneous modalities face significant challenges due to the curvature heterogeneity of parameter manifolds and mismatches in phase evolution. Traditional Euclidean optimization methods struggle to capture the complex interdependencies between heterogeneous modalities on non-Euclidean or geometrically inconsistent parameter manifolds. Furthermore, static alignment strategies often fail to suppress bifurcations and oscillatory behaviors in high-dimensional gradient flows, leading to unstable optimization trajectories across modalities. To address these issues, inspired by hyperbolic geometry and symplectic structures, this paper proposes the Hyperbolic Cosine-Based Symplectic Phase Alignment (HC-SPA) fusion optimization framework. The proposed approach leverages the geometric properties of hyperbolic space to coordinate gradient flows between modalities, aligns gradient update directions through a phase synchronization mechanism, and dynamically adjusts the optimization step size to adapt to manifold curvature. Experimental results on public fusion and semantic segmentation datasets demonstrate that HC-SPA significantly improves multimodal fusion performance and optimization stability, providing a new optimization perspective for complex multimodal tasks.