Unlabeled sample compression schemes and corner peelings for ample and maximum classes

We examine connections between combinatorial notions that arise in machine learning and topological cubical/simplicial geometry. These enable to export results from geometry learning. Our first main result is based on a geometric construction by Tracy Hall (2004) [20] of partial shelling the cross-polytope which can not be extended. From it, we derive maximum class VC dimension 3 without corners. This refutes several previous works In particular, it implies constructions optimal unlabeled sample compression schemes for classes are erroneous. On positive side present new an scheme classes. leave as open whether our extends ample classes, generalize Towards resolving this question, provide characterization terms unique sink orientations associated 1-inclusion graph.