This work studies embedding of arbitrary VC classes in well-behaved VC classes, focusing particularly on extremal classes. Our main result expresses an impossibility: such embeddings necessarily require a significant increase in dimension. In particular, we prove that for every $d$ there is a class with VC dimension $d$ that cannot be embedded in any extremal class of VC dimension smaller than exponential in $d$. In addition to its independent interest, this result has an important implication in learning theory, as it reveals a fundamental limitation of one of the most extensively studied approaches to tackling the long-standing sample compression conjecture. Concretely, the approach proposed by Floyd and Warmuth entails embedding any given VC class into an extremal class of a comparable dimension, and then applying an optimal sample compression scheme for extremal classes. However, our results imply that this strategy would in some cases result in a sample compression scheme at least exponentially larger than what is predicted by the sample compression conjecture. The above implications follow from a general result we prove: any extremal class with VC dimension $d$ has dual VC dimension at most $2d+1$. This bound is exponentially smaller than the classical bound $2^{d+1}-1$ of Assouad, which applies to general concept classes (and is known to be unimprovable for some classes). We in fact prove a stronger result, establishing that $2d+1$ upper bounds the dual Radon number of extremal classes. This theorem represents an abstraction of the classical Radon theorem for convex sets, extending its applicability to a wider combinatorial framework, without relying on the specifics of Euclidean convexity. The proof utilizes the topological method and is primarily based on variants of the Topological Radon Theorem.

翻译：本研究探讨了将任意VC类嵌入到性质良好的VC类中的问题，尤其关注极值类。我们的主要结果揭示了一种不可能性：此类嵌入必然导致维度的显著增加。具体而言，我们证明对于每个$d$，都存在一个VC维为$d$的类，其无法嵌入到任何VC维小于$d$的指数级的极值类中。除了其独立的理论意义外，这一结果对学习理论具有重要启示，因为它揭示了解决长期存在的样本压缩猜想的最广泛研究途径之一存在根本性局限。具体而言，Floyd和Warmuth提出的方法需要将任意给定的VC类嵌入到具有可比维度的极值类中，然后应用极值类的最优样本压缩方案。然而，我们的结果表明，该策略在某些情况下会导致样本压缩方案的规模至少是指数级大于样本压缩猜想所预测的规模。上述推论源于我们证明的一个一般性结果：任何VC维为$d$的极值类，其对偶VC维至多为$2d+1$。该界限指数级地小于Assouad提出的经典界限$2^{d+1}-1$（该界限适用于一般概念类，且已知对某些类是不可改进的）。我们实际上证明了一个更强的结果，确认$2d+1$是极值类对偶Radon数的上界。该定理代表了凸集经典Radon定理的抽象化，将其适用性扩展至更广泛的组合框架，而不依赖于欧几里得凸性的具体细节。证明采用了拓扑方法，主要基于拓扑Radon定理的变体。