摘要翻译:
考虑一类$\mh$二元函数$h:x\to\\{-1,+1\}$在有限区间$x=[0,B]\子集\real$上。将有限子集(Asample)$S\子集x$上$h$的{\em sample width}定义为$\w_s(h)\equiv\min_{x\in S}\w_h(x)$,其中$\w_h(x)=h(x)\max\{a\geq0:h(z)=h(x),x-a\leq z\leq x+a\}$。设$\mathbb{S}_\ell$是基数$\ell$的$x$中所有样本的空间,并考虑宽样本集,即{\em hypersets},其定义为$a_{\beta,h}=\{S\in\mathbb{S}_\ell:\w_{S}(h)\geq\beta\}$。通过应用Sauer-Shelah关于集合密度的结果,得到了类$\{A_{\\beta,h}:h\in\mh}$,$\beta>0$的增长函数(或迹)的一个上估计,即所有超序与基数为$m$的固定样本集$S\in\mathbb{S}\ell$相交所得到的可能二分性的个数。估计值为$2\sum_{i=0}^{2\l楼层B/(2\beta)\r楼层}{m-\ell\choose i}$。
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英文标题:
《On the Complexity of Binary Samples》
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作者:
Joel Ratsaby
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最新提交年份:
2008
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分类信息:
一级分类:Computer Science 计算机科学
二级分类:Discrete Mathematics 离散数学
分类描述:Covers combinatorics, graph theory, applications of probability. Roughly includes material in ACM Subject Classes G.2 and G.3.
涵盖组合学,图论,概率论的应用。大致包括ACM学科课程G.2和G.3中的材料。
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一级分类:Computer Science 计算机科学
二级分类:Artificial Intelligence 人工智能
分类描述:Covers all areas of AI except Vision, Robotics, Machine Learning, Multiagent Systems, and Computation and Language (Natural Language Processing), which have separate subject areas. In particular, includes Expert Systems, Theorem Proving (although this may overlap with Logic in Computer Science), Knowledge Representation, Planning, and Uncertainty in AI. Roughly includes material in ACM Subject Classes I.2.0, I.2.1, I.2.3, I.2.4, I.2.8, and I.2.11.
涵盖了人工智能的所有领域,除了视觉、机器人、机器学习、多智能体系统以及计算和语言(自然语言处理),这些领域有独立的学科领域。特别地,包括专家系统,定理证明(尽管这可能与计算机科学中的逻辑重叠),知识表示,规划,和人工智能中的不确定性。大致包括ACM学科类I.2.0、I.2.1、I.2.3、I.2.4、I.2.8和I.2.11中的材料。
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一级分类:Computer Science 计算机科学
二级分类:Machine Learning 机器学习
分类描述:Papers on all aspects of machine learning research (supervised, unsupervised, reinforcement learning, bandit problems, and so on) including also robustness, explanation, fairness, and methodology. cs.LG is also an appropriate primary category for applications of machine learning methods.
关于机器学习研究的所有方面的论文(有监督的,无监督的,强化学习,强盗问题,等等),包括健壮性,解释性,公平性和方法论。对于机器学习方法的应用,CS.LG也是一个合适的主要类别。
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英文摘要:
Consider a class $\mH$ of binary functions $h: X\to\{-1, +1\}$ on a finite interval $X=[0, B]\subset \Real$. Define the {\em sample width} of $h$ on a finite subset (a sample) $S\subset X$ as $\w_S(h) \equiv \min_{x\in S} |\w_h(x)|$, where $\w_h(x) = h(x) \max\{a\geq 0: h(z)=h(x), x-a\leq z\leq x+a\}$. Let $\mathbb{S}_\ell$ be the space of all samples in $X$ of cardinality $\ell$ and consider sets of wide samples, i.e., {\em hypersets} which are defined as $A_{\beta, h} = \{S\in \mathbb{S}_\ell: \w_{S}(h) \geq \beta\}$. Through an application of the Sauer-Shelah result on the density of sets an upper estimate is obtained on the growth function (or trace) of the class $\{A_{\beta, h}: h\in\mH\}$, $\beta>0$, i.e., on the number of possible dichotomies obtained by intersecting all hypersets with a fixed collection of samples $S\in\mathbb{S}_\ell$ of cardinality $m$. The estimate is $2\sum_{i=0}^{2\lfloor B/(2\beta)\rfloor}{m-\ell\choose i}$.
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PDF链接:
https://arxiv.org/pdf/0801.4794