度量空间中图嵌入的紧致且快速的泛化误差界
Tight and fast generalization error bound of graph embedding in metric space
AI总结：
本文针对非欧几里得度量空间中的图嵌入，通过局部Rademacher复杂度提出了更紧且更快的泛化误差上界，证明其性能优于现有理论预测，且所需训练数据更少。
摘要：
近期研究通过实验表明，我们可以在非欧几里得度量空间中实现有效且高效的图嵌入，其目标是在度量空间中获得反映图结构的顶点表示。具体而言，双曲空间中的图嵌入已在嵌入具有层次树状结构的数据（如自然语言、社交网络和知识库中的数据）方面取得了实验成功。然而，近期理论分析表明，非欧几里得图嵌入的泛化误差上界远高于欧几里得图嵌入的上界，其中高泛化误差意味着数据的不完整性和噪声会显著损害学习性能。这意味着现有界无法保证在实际训练数据规模下非欧几里得度量空间中图嵌入的成功，这可能阻碍非欧几里得图嵌入在真实问题中的应用。本文通过将模型的局部Rademacher复杂度作为表示对距离的函数集进行评估，提供了图嵌入泛化误差的一个新上界。我们的界阐明，非欧几里得度量空间（包括双曲空间）中图嵌入的性能优于现有上界所暗示的性能。具体而言，我们的新上界关于度量空间的几何半径$R$是多项式的，并且最快可达$O(\frac{1}{S})$，其中$S$是训练数据规模。我们的界显著比现有界更紧且更快，现有界关于$R$可能是指数级的，且最快为$O(\frac{1}{\sqrt{S}})$。对示例情况的具体计算表明，非欧几里得度量空间中的图嵌入可以在比现有界所建议的所需训练数据小得多的情况下优于欧几里得空间中的图嵌入。
英文摘要：
Recent studies have experimentally shown that we can achieve in non-Euclidean metric space effective and efficient graph embedding, which aims to obtain the vertices' representations reflecting the graph's structure in the metric space. Specifically, graph embedding in hyperbolic space has experimentally succeeded in embedding graphs with hierarchical-tree structure, e.g., data in natural languages, social networks, and knowledge bases. However, recent theoretical analyses have shown a much higher upper bound on non-Euclidean graph embedding's generalization error than Euclidean one's, where a high generalization error indicates that the incompleteness and noise in the data can significantly damage learning performance. It implies that the existing bound cannot guarantee the success of graph embedding in non-Euclidean metric space in a practical training data size, which can prevent non-Euclidean graph embedding's application in real problems. This paper provides a novel upper bound of graph embedding's generalization error by evaluating the local Rademacher complexity of the model as a function set of the distances of representation couples. Our bound clarifies that the performance of graph embedding in non-Euclidean metric space, including hyperbolic space, is better than the existing upper bounds suggest. Specifically, our new upper bound is polynomial 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 bound is significantly tighter and faster than the existing one, which can be exponential to $R$ and $O(\frac{1}{\sqrt{S}})$ at the fastest. Specific calculations on example cases show that graph embedding in non-Euclidean metric space can outperform that in Euclidean space with much smaller training data than the existing bound has suggested.
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