基于嵌入Laplacian离散的多尺度图比较
Multiscale Graph Comparison via the Embedded Laplacian Discrepancy
摘要：
Laplacian特征向量捕获图上的自然社区结构，并广泛用于谱聚类和流形学习。然而，使用拉普拉斯特征向量作为多尺度图比较的嵌入的目的是有限的。在这里，我们提出了嵌入式拉普拉斯离散（ELD）作为一个简单而快速的方法来比较图（潜在的不同大小）的基础上的相似性图的社区结构。ELD通过将图表示为常见的低维空间中的点云来操作，在该低维空间上可以有效地计算自然的基于Wasserstein的距离。通过任何基于特征向量的方法比较图的一个主要挑战是由于符号翻转和基础对称性而可能出现的潜在模糊性。ELD利用一个简单的对称化技巧来绕过任何符号歧义。对于比较图，没有任何歧义，由于基础对称性（即频谱是简单的），我们表明，ELD成为一个自然的伪度量，享有良好的性能，如图同构下的不变性。对于比较图与非简单的频谱，我们提出了一个程序，通过一个简单的扰动技术来解决任何模糊的基础对称性近似ELD。我们表明，这种扰动是稳定的，使用矩阵扰动理论在温和的假设下，是直接在实践中验证。我们证明了优秀的适用性的ELD方法在模拟和真实的数据集。
英文摘要：
Laplacian eigenvectors capture natural community structures on graphs and are widely used in spectral clustering and manifold learning. The use of Laplacian eigenvectors as embeddings for the purpose of multiscale graph comparison has however been limited. Here we propose the Embedded Laplacian Discrepancy (ELD) as a simple and fast approach to compare graphs (of potentially different sizes) based on the similarity of the graphs' community structures. The ELD operates by representing graphs as point clouds in a common, low-dimensional space, on which a natural Wasserstein-based distance can be efficiently computed. A main challenge in comparing graphs through any eigenvector-based approaches is the potential ambiguity that could arise due to sign-flips and basis symmetries. The ELD leverages a simple symmetrization trick to bypass any sign ambiguities. For comparing graphs that do not have any ambiguities due to basis symmetries (i.e. the spectrums are simple), we show that the ELD becomes a natural pseudo-metric that enjoys nice properties such as invariance under graph isomorphism. For comparing graphs with non-simple spectrums, we propose a procedure to approximate the ELD via a simple perturbation technique to resolve any ambiguity from basis symmetries. We show that such perturbations are stable using matrix perturbation theory under mild assumptions that are straightforward to verify in practice. We demonstrate the excellent applicability of the ELD approach on both simulated and real datasets.
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