We address the problem of setting the kernel bandwidth, epps, used by Manifold Learning algorithms to construct the graph Laplacian. Exploiting the connection between manifold geometry, represented by the Riemannian metric, and the Laplace-Beltrami operator, we set epps by optimizing the Laplacian's ability to preserve the geometry of the data.

Forl = 2, i. e., for 3 - cycles, thisrelationship is X ikX kj= X ij. Figure 1: Averagematchingerrorunderthe LBCmodel ( left) and LACmodel ( right).

Interestingly, and in a sharp departure from previous proposals, this operation on coarsened graphs is often oriented, even when the original graph is undirected.

H . (32) We refer to ( H

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