  Manifold learning methods play a prominent role in nonlinear dimensionality
reduction and other tasks involving high-dimensional data sets with low
intrinsic dimensionality. Many of these methods are graph-based: they associate
a vertex with each data point and a weighted edge with each pair. Existing
theory shows that the Laplacian matrix of the graph converges to the
Laplace-Beltrami operator of the data manifold, under the assumption that the
pairwise affinities are based on the Euclidean norm. In this paper, we
determine the limiting differential operator for graph Laplacians constructed
using $\textit{any}$ norm. Our proof involves an interplay between the second
fundamental form of the manifold and the convex geometry of the given norm's
unit ball. To demonstrate the potential benefits of non-Euclidean norms in
manifold learning, we consider the task of mapping the motion of large
molecules with continuous variability. In a numerical simulation we show that a
modified Laplacian eigenmaps algorithm, based on the Earthmover's distance,
outperforms the classic Euclidean Laplacian eigenmaps, both in terms of
computational cost and the sample size needed to recover the intrinsic
geometry.
