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Geodesic distance matrices can reveal shape properties that are largelyinvariant to non-rigid deformations, and thus are often used to analyze andrepresent 3-D shapes. However, these matrices grow quadratically with thenumber of points. Thus for large point sets it is common to use a low-rankapproximation to the distance matrix, which fits in memory and can beefficiently analyzed using methods such as multidimensional scaling (MDS). Inthis paper we present a novel sparse method for efficiently representinggeodesic distance matrices using biharmonic interpolation. This method exploitsknowledge of the data manifold to learn a sparse interpolation operator thatapproximates distances using a subset of points. We show that our method is 2xfaster and uses 20x less memory than current leading methods for solving MDS onlarge point sets, with similar quality. This enables analyses of large pointsets that were previously infeasible.

Efficient, sparse representation of manifold distance matrices for classical scaling