最优多值粉碎
Optimal Multivalued Shattering
摘要：
我们已经找到了最一般的扩展著名的绍尔，Perles和Shelah，Vapnik和Chervonenkis结果从0-1序列$k$ -ary码仍然给一个多项式的限制。令$\mathcal{C}\subseteq \{0,1,..., k-1}^n$是长度为$n$的$k$元码。对于坐标$S\subset{1,2,...,n}$的子集，$\mathcal{C}$到$S$的投影由$\mathcal{C}|_S$表示。如果$\mathcal{C}|_S$包含具有坐标$i$和$j$的所有$2^{|S|}$不同向量（码字），则我们称$\mathcal{C}$ $(i,j)$ -{ \em shatters} $S$。假设$\mathcal{C}$不对每个$1\leq i< j\leq q$分解任何大小为$s_{i,j}\geq 1$的坐标集，并设$p=\sum (s_{i,j}-1)$。使用自然归纳法，我们证明了$$ |{\mathcal C}|\leq O(n^p)$$为任何给定的$p$为$n\to \infty$，并给出了一个建设表明，这个指数是最好的可能。几个开放的问题被提到。
英文摘要：
We have found the most general extension of the celebrated Sauer, Perles and Shelah, Vapnik and Chervonenkis result from 0-1 sequences to $k$-ary codes still giving a polynomial bound. Let $\mathcal{C}\subseteq \{0,1,..., k-1}^n$ be a $k$-ary code of length $n$. For a subset of coordinates $S\subset{1,2,...,n}$ the projection of $\mathcal{C}$ to $S$ is denoted by $\mathcal{C}|_S$. We say that $\mathcal{C}$ $(i,j)$-{\em shatters} $S$ if $\mathcal{C}|_S$ contains all the $2^{|S|}$ distinct vectors (codewords) with coordinates $i$ and $j$. Suppose that $\mathcal{C}$ does not $(i,j)$-shatter any coordinate set of size $s_{i,j}\geq 1$ for every $1\leq i< j\leq q$ and let $p=\sum (s_{i,j}-1)$. Using a natural induction we prove that $$ |{\mathcal C}|\leq O(n^p)$$ for any given $p$ as $n\to \infty$ and give a construction showing that this exponent is the best possible. Several open problems are mentioned.
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