Spectral clustering is an important graph clustering technique. However, it suffers from a scalability problem due to the involved computationally expensive eigen-decomposition procedure and is highly sensitive to noisy nodes in the graph. In this work we solve the two problems simultaneously by using spectrum-preserving node aggregation to generate a nearly-noiseless concise representation of the data set. Specifically, we create a small set of spectral-representative pseudo-nodes based on spectral analysis. Then, standard spectral clustering algorithm is performed on the smaller node set. Finally, we map the clusters of pseudo nodes back to find the cluster memberships of the data points in the original data set. The proposed framework has a nearly-linear time complexity. Meanwhile, the clustering accuracy can be significantly improved. The experimental results show dramatically improved clustering performance when compared with state-of-the-art methods.

Graph learning plays an important role in many data mining and machine learning tasks, such as manifold learning, data representation and analysis, dimensionality reduction, data clustering, and visualization, etc. In this work, we introduce a highly-scalable spectral graph densification approach (GRASPEL) for graph topology learning from data. By limiting the precision matrix to be a graph-Laplacian-like matrix, our approach aims to learn sparse undirected graphs from potentially high-dimensional input data. A very unique property of the graphs learned by GRASPEL is that the spectral embedding (or approximate effective-resistance) distances on the graph will encode the similarities between the original input data points. By leveraging high-performance spectral methods, sparse yet spectrally-robust graphs can be learned by identifying and including the most spectrally-critical edges into the graph. Compared with prior state-of-the-art graph learning approaches, GRASPEL is more scalable and allows substantially improving computing efficiency and solution quality of a variety of data mining and machine learning applications, such as manifold learning, spectral clustering (SC), and dimensionality reduction (DR).

Algebraic multigrid (AMG) is a class of high-performance linear solvers based on multigrid principles. Compared to geometric multigrid (GMG) solvers that rely on the geometric information of underlying problems, AMG solvers build hierarchical coarse level problems according to the input matrices. Graph-theoretic Algebraic Multigrid (AMG) algorithms have emerged for solving large Symmetric Diagonally Dominant (SDD) matrices by taking advantages of spectral properties of graph Laplacians. This paper proposes a Sparsified graph-theoretic Algebraic Multigrid (SAMG) framework that allows efficiently constructing nearly-linear sized graph Laplacians for coarse level problems while maintaining good spectral approximation during the AMG setup phase by leveraging a scalable spectral graph sparsification engine. Our experimental results show that the proposed method can offer more scalable performance than existing graph-theoretic AMG solvers for solving large SDD matrices in integrated circuit (IC) simulations, 3D-IC thermal analysis, image processing, finite element analysis as well as data mining and machine learning applications.