Ranking and Sparsifying a Connection Graph

Abstract Many problems arising in dealing with high-dimensional data sets involve connection graphs in which each edge is associated with both an edge weight and a d-dimensional linear transformation. We consider vectorized versions of PageRank and effective resistance that can be used as basic tools for organizing and analyzing complex data sets. For example, generalized PageRank and effective resistance can be utilized to derive and modify diffusion distances for vector diffusion maps in data and image processing. Furthermore, the edge-ranking of the connection graphs determined by vectorized PageRank and effective resistance are an essential part of sparsification algorithms that simplify and preserve the global structure of connection graphs. In addition, we examine consistencies in a connection graph, particularly in the applications of recovering low-dimensional data sets and the reduction of noise. In these applications, we analyze the effect of deleting edges with high edge rank.