Prior constraints are imposed upon a learning problem in the form of distance measures. Prototypical 2-D point sets and graphs are learned by clustering with point matching and graph matching dis(cid:173) tance measures. The point matching distance measure is approx. It operates between noisy images with missing and spurious points. The graph matching distance measure operates on weighted graphs and is invariant under per(cid:173) mutations.

Recently, the importance of such preknowledge for learning has been convincingly argued from a statistical framework [Geman et al., 1992]. Researchers have proposed that our brains may incorporate preknowledge in the form of distance measures [Shepard, 1989]. The neural network community has begun to explore this idea via tangent distance [Simard et al., 1993], model learning [Williams et al., 1993] and point matching distances [Gold et al., 1994]. However, only the point matching distances have been invariant under permutations. Here we extend that work by enhancing both the scope and function of those distance measures, significantly expanding the problem domains where learning may take place. We learn objects consisting of noisy 2-D point-sets or noisy weighted graphs by clustering with point matching and graph matching distance measures. The point matching measure is approx.

Learning is formulated as an optimization problem .