This paper develops an approximation to the (effective) $p$-resistance and applies it to multi-class clustering. Spectral methods based on the graph Laplacian and its generalization to the graph $p$-Laplacian have been a backbone of non-euclidean clustering techniques. The advantage of the $p$-Laplacian is that the parameter $p$ induces a controllable bias on cluster structure. The drawback of $p$-Laplacian eigenvector based methods is that the third and higher eigenvectors are difficult to compute. Thus, instead, we are motivated to use the $p$-resistance induced by the $p$-Laplacian for clustering. For $p$-resistance, small $p$ biases towards clusters with high internal connectivity while large $p$ biases towards clusters of small "extent," that is a preference for smaller shortest-path distances between vertices in the cluster. However, the $p$-resistance is expensive to compute. We overcome this by developing an approximation to the $p$-resistance. We prove upper and lower bounds on this approximation and observe that it is exact when the graph is a tree. We also provide theoretical justification for the use of $p$-resistance for clustering. Finally, we provide experiments comparing our approximated $p$-resistance clustering to other $p$-Laplacian based methods.

翻译：本文提出了对（有效）$p$-电阻的近似方法，并将其应用于多类聚类问题。基于图拉普拉斯算子及其推广到图$p$-拉普拉斯算子的谱方法，一直是非欧几里得聚类技术的核心。$p$-拉普拉斯算子的优势在于参数$p$对聚类结构引入可控的偏置。然而，基于$p$-拉普拉斯算子特征向量的方法存在计算其三阶及以上高阶特征向量困难的问题。因此，我们转而利用由$p$-拉普拉斯算子诱导的$p$-电阻进行聚类。对于$p$-电阻，较小$p$值倾向于内部连接更紧密的聚类，而较大$p$值则倾向于"范围"较小的聚类，即偏好聚类内顶点间最短路径距离更小的结构。由于$p$-电阻计算成本高昂，我们通过开发其近似方法克服了这一瓶颈。我们证明了该近似方法的上下界，并观察到当图为树结构时近似结果精确。同时，我们为使用$p$-电阻进行聚类提供了理论依据。最后，通过实验将本文提出的近似$p$-电阻聚类方法与其他基于$p$-拉普拉斯算子的方法进行了对比。