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Recent studies have experimentally shown that we can achieve in non-Euclideanmetric space effective and efficient graph embedding, which aims to obtain thevertices' representations reflecting the graph's structure in the metric space.Specifically, graph embedding in hyperbolic space has experimentally succeededin embedding graphs with hierarchical-tree structure, e.g., data in naturallanguages, social networks, and knowledge bases. However, recent theoreticalanalyses have shown a much higher upper bound on non-Euclidean graphembedding's generalization error than Euclidean one's, where a highgeneralization error indicates that the incompleteness and noise in the datacan significantly damage learning performance. It implies that the existingbound cannot guarantee the success of graph embedding in non-Euclidean metricspace in a practical training data size, which can prevent non-Euclidean graphembedding's application in real problems. This paper provides a novel upperbound of graph embedding's generalization error by evaluating the localRademacher complexity of the model as a function set of the distances ofrepresentation couples. Our bound clarifies that the performance of graphembedding in non-Euclidean metric space, including hyperbolic space, is betterthan the existing upper bounds suggest. Specifically, our new upper bound ispolynomial in the metric space's geometric radius $R$ and can be$O(\frac{1}{S})$ at the fastest, where $S$ is the training data size. Our boundis significantly tighter and faster than the existing one, which can beexponential to $R$ and $O(\frac{1}{\sqrt{S}})$ at the fastest. Specificcalculations on example cases show that graph embedding in non-Euclidean metricspace can outperform that in Euclidean space with much smaller training datathan the existing bound has suggested.

Tight and fast generalization error bound of graph embedding in metric space