Classical Scaling Revisited

Multidimensional-scaling (MDS) is an information analysis tool. It involves the evaluation of distances between data points, which is a quadratic space-time problem. Then, MDS procedures find an embedding of the points in a low dimensional Euclidean (flat) domain, optimizing for the similarity of inter-points distances. We present an efficient solver for Classical Scaling (a specific MDS model) by extending the distances measured from a subset of the points to the rest, while exploiting the smoothness property of the distance functions. The smoothness is measured by the L2 norm of the Laplace-Beltrami operator applied to the unknown distance function. The Laplace Beltrami reflects the local differential relations between points, and can be computed in linear time. Classical-scaling is thereby reformulated into a quasi-linear space-time complexities procedure.