We examine connections between combinatorial notions that arise in machine learning and topological notions in cubical/simplicial geometry. These connections enable to export results from geometry to machine learning. Our first main result is based on a geometric construction by Tracy Hall (2004) of a partial shelling of the cross-polytope which can not be extended. We use it to derive a maximum class of VC dimension 3 that has no corners. This refutes several previous works in machine learning from the past 11 years. In particular, it implies that all previous constructions of optimal unlabeled sample compression schemes for maximum classes are erroneous. On the positive side we present a new construction of an unlabeled sample compression scheme for maximum classes. We leave as open whether our unlabeled sample compression scheme extends to ample (a.k.a. lopsided or extremal) classes, which represent a natural and far-reaching generalization of maximum classes. Towards resolving this question, we provide a geometric characterization in terms of unique sink orientations of the 1-skeletons of associated cubical complexes.

翻译：我们检查机器学习中产生的组合概念与立体/模拟几何学中的地形概念之间的联系。 这些连接能够导出几何结果到机器学习。 我们的第一个主要结果是基于Tracy Hall(2004年)对无法延伸的交叉流体进行部分炮击的几何构造(2004年), 我们用它来得出一个最高VC维度3的无角最大等级。 这驳斥了过去11年来在机器中学习的数项工程。 特别是, 它意味着所有以前为最高等级构建的最佳无标签样本压缩计划都是错误的。 在正面, 我们提出了一个新的最高等级未贴标签的样本压缩计划。 我们让未贴标签的样本压缩计划是否扩大到足够( a.k.a.loply或extremal) 的( a.k.a.loply或extremal) 班级, 它代表最高等级的自然和深远的概括。 为了解决这个问题, 我们提供了一个相关立方复合体1号单箱独特的水槽方向的几何测量特征。