Graph Learning via Spectral Densification

Graph learning plays 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. For the first time, we present a highly-scalable spectral graph densification approach (GRASPEL) for graph learning from data. By limiting the precision matrix to be a graph-Laplacian-like matrix in graphical Lasso, our approach aims to learn ultra-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 interleaving the latest high-performance nearly-linear time spectral methods, ultrasparse 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.