We show that the topes of a complex of oriented matroids (abbreviated COM) of VC-dimension $d$ admit a proper labeled sample compression scheme of size $d$. This considerably extends results of Moran and Warmuth on ample classes, of Ben-David and Litman on affine arrangements of hyperplanes, and of the authors on complexes of uniform oriented matroids, and is a step towards the sample compression conjecture -- one of the oldest open problems in computational learning theory. On the one hand, our approach exploits the rich combinatorial cell structure of COMs via oriented matroid theory. On the other hand, viewing tope graphs of COMs as partial cubes creates a fruitful link to metric graph theory.

翻译：我们证明，VC维数为$d$的有向拟阵复合形（简称COM）的顶位类（tope）存在一个规模为$d$的恰当标记样本压缩方案。这一结果显著扩展了Moran和Warmuth关于充足类（ample classes）的结论、Ben-David和Litman关于超平面仿射排列的研究，以及作者团队关于一致有向拟阵复合形的工作，并朝着样本压缩猜想——计算学习理论中最古老的开放问题之一——迈出了重要一步。一方面，我们的方法通过有向拟阵理论充分利用了COM丰富的组合胞结构；另一方面，将COM的顶位图视为部分立方体（partial cubes）为度量图论建立了富有成效的联系。