Knowledge graph embedding (KGE) relies on the geometry of the embedding space to encode semantic and structural relations. Existing methods place all entities on one homogeneous manifold, Euclidean, spherical, hyperbolic, or their product/multi-curvature variants, to model linear, symmetric, or hierarchical patterns. Yet a predefined, homogeneous manifold cannot accommodate the sharply varying curvature that real-world graphs exhibit across local regions. Since this geometry is imposed a priori, any mismatch with the knowledge graph's local curvatures will distort distances between entities and hurt the expressiveness of the resulting KGE. To rectify this, we propose RicciKGE to have the KGE loss gradient coupled with local curvatures in an extended Ricci flow such that entity embeddings co-evolve dynamically with the underlying manifold geometry towards mutual adaptation. Theoretically, when the coupling coefficient is bounded and properly selected, we rigorously prove that i) all the edge-wise curvatures decay exponentially, meaning that the manifold is driven toward the Euclidean flatness; and ii) the KGE distances strictly converge to a global optimum, which indicates that geometric flattening and embedding optimization are promoting each other. Experimental improvements on link prediction and node classification benchmarks demonstrate RicciKGE's effectiveness in adapting to heterogeneous knowledge graph structures.

翻译：知识图谱嵌入（KGE）依赖于嵌入空间的几何结构来编码语义和结构关系。现有方法将所有实体置于单一均匀流形上，如欧几里得、球面、双曲空间或其乘积/多曲率变体，以建模线性、对称或层次化模式。然而，预定义的均匀流形无法适应现实世界图谱在局部区域表现出的急剧变化的曲率。由于这种几何结构是先验强加的，任何与知识图谱局部曲率的不匹配都会扭曲实体间的距离，损害最终KGE的表达能力。为纠正此问题，我们提出RicciKGE，将KGE损失梯度与局部曲率在扩展里奇流中耦合，使实体嵌入与底层流形几何结构通过协同演化实现动态相互适应。理论上，当耦合系数有界且适当选择时，我们严格证明：i) 所有边曲率呈指数衰减，意味着流形被驱动趋向欧几里得平坦性；ii) KGE距离严格收敛至全局最优解，表明几何平坦化与嵌入优化相互促进。在链接预测和节点分类基准测试上的实验改进证明了RicciKGE在适应异构知识图谱结构方面的有效性。