  We introduce a novel approach to measure the behavior of a geometric operator
before and after coarsening. By comparing eigenvectors of the input operator
and its coarsened counterpart, we can quantitatively and visually analyze how
well the spectral properties of the operator are maintained. Using this
measure, we show that standard mesh simplification and algebraic coarsening
techniques fail to maintain spectral properties. In response, we introduce a
novel approach for spectral coarsening. We show that it is possible to
significantly reduce the sampling density of an operator derived from a 3D
shape without affecting the low-frequency eigenvectors. By marrying techniques
developed within the algebraic multigrid and the functional maps literatures,
we successfully coarsen a variety of isotropic and anisotropic operators while
maintaining sparsity and positive semi-definiteness. We demonstrate the utility
of this approach for applications including operator-sensitive sampling, shape
matching, and graph pooling for convolutional neural networks.
