Playing Billiard in Version Space

A ray-tracing method inspired by ergodic billiards is used to estimate the theoretically best decision rule for a set of linear separable examples. While the Bayes-optimum requires a majority decision over all Perceptrons separating the example set, the problem considered here corresponds to finding the single Perceptron with best average generalization probability. For randomly distributed examples the billiard estimate agrees with known analytic results. In real-life classification problems the generalization error is consistently reduced compared to the maximal stability Perceptron.Comment: uuencoded, gzipped PostScript file, 127576 bytes To recover 1) save file as bayes.uue. Then 2) uudecode bayes.uue and 3) gunzip bayes.ps.g

Surface-area-minimizing n-hedral Tiles

We provide a list of conjectured surface-area-minimizing n-hedral tiles of space for n from 4 to 14, previously known only for n equal to 5 and 6. We find the optimal orientation-preserving tetrahedral tile (n=4), and we give a nice new proof for the optimal 5-hedron (a triangular prism)

Discrete Differential Geometry

This is the collection of extended abstracts for the 26 lectures and the open problem session at the fourth Oberwolfach workshop on Discrete Differential Geometry