Given a set $E \subset {\Bbb F}_q^3$, where ${\Bbb F}_q$ is the field with $q$ elements. Consider a set of "classifiers" ${\mathcal H}^3_t(E)=\{h_y: y \in E\}$, where $h_y(x)=1$ if $x \cdot y=t$, $x \in E$, and $0$ otherwise. We are going to prove that if $|E| \ge Cq^{\frac{11}{4}}$, with a sufficiently large constant $C>0$, then the Vapnik-Chervonenkis dimension of ${\mathcal H}^3_t(E)$ is equal to $3$. In particular, this means that for sufficiently large subsets of ${\Bbb F}_q^3$, the Vapnik-Chervonenkis dimension of ${\mathcal H}^3_t(E)$ is the same as the Vapnik-Chervonenkis dimension of ${\mathcal H}^3_t({\Bbb F}_q^3)$. In some sense the proof leads us to consider the most complicated possible configuration that can always be embedded in subsets of ${\Bbb F}_q^3$ of size $\ge Cq^{\frac{11}{4}}$.
We study combinatorial complexity of certain classes of products of intervals in $\mathbb{R}^d$, from the point of view of Vapnik-Chervonenkis geometry. As a consequence of the obtained results, we conclude that the Vapnik-Chervonenkis dimension of the set of balls in $\ell_\infty^d$ -- which denotes $\R^d$ equipped with the sup norm -- equals $\lfloor (3d+1)/2\rfloor$.
The Vapnik-Chervonenkis (VC) dimension of a collection of subsets of a set is an important combinatorial concept in settings such as discrete geometry and machine learning. In this paper we prove that the VC dimension of the family of $d$-dimensional cubes in $\mathbb R^d$ is $\lfloor(3d+1)/2\rfloor$.
Shattering-extremal set systems of small VC-dimension
如果$2^S={F \cap S : F \in \mathcal{F}}$，我们说集合系统$\mathcal{F}\subseteq 2^{[n]}$粉碎给定的集合$S\subseteq [n]$。Sauer不等式指出，一般来说，集合系统$\mathcal{F}$至少粉碎$|\mathcal{F}|$集合。这里我们集中讨论平等的问题。一个集合系统被称为破碎极值，如果它正好破碎$|\mathcal{F}|$集。我们刻画了Vapnik-Chervonenkis维数为1的破碎极值集系统的包含图。此外，从极值的角度来看，我们涉及到有界Vapnik-Chervonenkis维的集合系统，他们的投影。
Shattering-extremal set systems of VC dimension at most 2
如果$2^S=\{F \cap S : F \in \mathcal{F}\}$，我们说集合系统$\mathcal{F}\subseteq 2^{[n]}$粉碎给定的集合$S\subseteq [n]$。Sauer不等式指出，一般来说，集合系统$\mathcal{F}$至少粉碎$|\mathcal{F}|$集合。在这里，我们集中讨论平等的问题。一个集合系统被称为破碎极值，如果它正好破碎$|\mathcal{F}|$集。本文利用包含图刻画了Vapnik-Chervonenkis维数$2$的破碎极集系统，并作为推论回答了\cite{VC1}关于在Vapnik-Chervonenkis维数$2$族的情况下从破碎极集系统中去掉元素的一个公开问题.
VC-dimension and VC-density are measures of combinatorial complexity of set systems. VC-dimension was first introduced in the context of statistical learning theory, and is tightly related to the sample complexity in PAC learning. VC-density is a refinement of VC-dimension. Both notions are also studied in model theory, in the context of \emph{dependent} theories. A set system that is definable by a formula of first-order logic with parameters has finite VC-dimension if and only if the formula is a dependent formula. In this paper we study the VC-dimension and the VC-density of the edge relati…
Let $X$ be a set and ${\mathcal H}$ a collection of functions from $X$ to $\{0,1\}$. We say that ${\mathcal H}$ shatters a finite set $C \subset X$ if the restriction of ${\mathcal H}$ yields every possible function from $C$ to $\{0,1\}$. The VC-dimension of ${\mathcal H}$ is the largest number $d$ such that there exists a set of size $d$ shattered by ${\mathcal H}$, and no set of size $d+1$ is shattered by ${\mathcal H}$. Vapnik and Chervonenkis introduced this idea in the early 70s in the context of learning theory, and this idea has also had a significant impact on other areas of mathematic…
Given a family $\mathcal F$ of $k$-element sets, $S_1,\ldots,S_r\in\mathcal F$ form an {\em $r$-sunflower} if $S_i \cap S_j =S_{i'} \cap S_{j'}$ for all $i \neq j$ and $i' \neq j'$. According to a famous conjecture of Erd\H os and Rado (1960), there is a constant $c=c(r)$ such that if $|\mathcal F|\ge c^k$, then $\mathcal F$ contains an $r$-sunflower. We come close to proving this conjecture for families of bounded {\em Vapnik-Chervonenkis dimension}, VC-dim$(\mathcal F)\le d$. In this case, we show that $r$-sunflowers exist under the slightly stronger assumption $|\mathcal F|\ge2^{10k(dr)^{2\…
Let $\mathcal S$ be a family of subsets of a set $X$ of cardinality $m$ and $\text{VC-dim}(\mathcal S)$ be the Vapnik-Chervonenkis dimension of $\mathcal S$. Haussler, Littlestone, and Warmuth (Inf. Comput., 1994) proved that if $G_1(\mathcal S)=(V,E)$ is the subgraph of the hypercube $Q_m$ induced by $\mathcal S$ (called the 1-inclusion graph of $\mathcal S$), then $\frac{|E|}{|V|}\le \text{VC-dim}({\mathcal S})$. Haussler (J. Combin. Th. A, 1995) presented an elegant proof of this inequality using the shifting operation. In this note, we adapt the shifting technique to prove that if $\mathca…
我们研究的复杂性，避免某些子结构的集系统（超图）的命中集问题。特别是，我们描述了经典的和参数化的复杂性的问题时，Vapnik-Chervonenkis维（VC维）的输入是小的。VC维是集合系统复杂性的一个自然度量。已知具有几何或图论风味的命中集的几个易处理的实例具有低VC维数。在有界VC维的集合系统中，已知命中集允许有效且几乎最优的近似算法（Brönnimann and Goodrich，1995; Even，Rawitz，and Shahar，2005; Agarwal and Pan，2014）。与这些近似结果相反，低VC维并不一定意味着参数化意义上的易处理性。事实上，我们证明了在VC维为2的输入上，即使对偶集系统的VC维也是2，命中集也是W[1]-硬的。因此，即使在这种可以说是简单的情况下，命中集也不太可能是固定参数的。这回答了King在2010年提出的一个公开问题。对于VC维数为1的集合系统，证明了击中集在多项式时间内可解。为了弥合VC维1和2的输入类别之间的复杂性差距，我们使用了比VC维更细粒度的度量。根据这一措施，我们确定了一个尖锐的阈值，击中集的复杂性从多项式时间可解过渡到NP难。恰好位于阈值之下的易处理类是边覆盖的推广，从而扩展了命中集的多项式时间易处理性的域。
