  Can one reduce the size of a graph without significantly altering its basic
properties? The graph reduction problem is hereby approached from the
perspective of restricted spectral approximation, a modification of the
spectral similarity measure used for graph sparsification. This choice is
motivated by the observation that restricted approximation carries strong
spectral and cut guarantees, and that it implies approximation results for
unsupervised learning problems relying on spectral embeddings.
  The paper then focuses on coarsening---the most common type of graph
reduction. Sufficient conditions are derived for a small graph to approximate a
larger one in the sense of restricted similarity. These findings give rise to
nearly-linear algorithms that, compared to both standard and advanced graph
reduction methods, find coarse graphs of improved quality, often by a large
margin, without sacrificing speed.
