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Spectral compression of the geometry of triangle meshes achievesgood results in practice, but there has been little or notheoretical support for the optimality of this compression. We showthat, for certain classes of geometric mesh models, spectraldecomposition using the eigenvectors of the symmetric Laplacian ofthe connectivity graph is equivalent to principal componentanalysis on that class, when equipped with a natural probabilitydistribution. Our proof treats connected one-and two-dimensionalmeshes with fixed convex boundaries, and is based on an asymptoticapproximation of the probability distribution in thetwo-dimensional case. The key component of the proof is that theLaplacian is identical, up to a constant factor, to the inversecovariance matrix of the distribution of valid mesh geometries.Hence, spectral compression is optimal, in the mean square errorsense, for these classes of meshes under some natural assumptionson their distribution.