Optimal Multivalued Shattering

TL;DR

This paper generalizes classical VC-dimension bounds from binary to multivalued codes, establishing polynomial bounds for the size of codes that avoid certain multicolored shattering configurations.

Contribution

It extends Sauer's lemma to multivalued codes with multiple symbols, providing tight polynomial bounds on code size based on shattering restrictions.

Findings

Established polynomial bounds for multivalued codes avoiding specific shattering patterns.

Proved the bounds are tight with matching constructions.

Extended classical VC theory to a broader multivalued setting.

Abstract

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.