Title: GRASPEL: GRAPH SPECTRAL LEARNING AT SCALE. Abstract: Learning meaningful graphs from data plays important roles in many data mining and machine learning tasks, such as data representation and analysis, dimension reduction, data clustering, and visualization, etc. In this work, we present a  scalable spectral approach to graph learning from data. By limiting the precision matrix to be a graph Laplacian, our approach aims to estimate ultra-sparse weighted graphs and has a clear connection with the prior graphical Lasso method. By interleaving nearly-linear time spectral graph sparsification,  coarsening and embedding procedures, ultra-sparse yet spectrally-stable graphs can be iteratively constructed in a highly-scalable manner. Compared with prior graph learning approaches that do not scale to large problems, our approach is highly-scalable for constructing graphs that can immediately lead to substantially improved computing efficiency and solution quality for a variety of data mining and machine learning applications, such as spectral clustering (SC), and t-Distributed Stochastic Neighbor Embedding (t-SNE).  