While orthogonalization exists in current dimensionality reduction methods in spectral clustering on undirected graphs, it does not scale in parallel computing environments. We propose four orthogonalization-free methods for spectral clustering. Our methods optimize one of two objective functions with no spurious local minima. In theory, two methods converge to features isomorphic to the eigenvectors corresponding to the smallest eigenvalues of the symmetric normalized Laplacian. The other two converge to features isomorphic to weighted eigenvectors weighting by the square roots of eigenvalues. We provide numerical evidence on the synthetic graphs from the IEEE HPEC Graph Challenge to demonstrate the effectiveness of the orthogonalization-free methods. Numerical results on the streaming graphs show that the orthogonalization-free methods are competitive in the streaming graph scenario since they can take full advantage of the computed features of previous graphs and converge fast. Our methods are also more scalable in parallel computing environments because orthogonalization is unnecessary. Numerical results are provided to demonstrate the scalability of our methods. Consequently, our methods have advantages over other dimensionality reduction methods when handling spectral clustering for large streaming graphs.

翻译：尽管当前无向图谱聚类中的降维方法存在正交化过程，但其在并行计算环境中无法有效扩展。本文提出了四种免正交化谱聚类方法。这些方法优化了两个目标函数之一，且不存在伪局部极小值。理论上，其中两种方法收敛到与对称归一化拉普拉斯矩阵最小特征值对应特征向量同构的特征；另外两种方法收敛到按特征值平方根加权的加权特征向量同构的特征。我们在IEEE HPEC图挑战赛的合成图上提供了数值证据，证明了免正交化方法的有效性。在流图上的数值结果表明，免正交化方法在流图场景中具有竞争力，因为它们能充分利用先前图计算得到的特征并快速收敛。由于无需正交化，我们的方法在并行计算环境中也更具可扩展性。提供的数值结果验证了我们方法的可扩展性。因此，在处理大规模流图谱聚类时，我们的方法相较于其他降维方法具有显著优势。