布尔函数的最近邻表示中的计数与覆盖
Counting and Covering in Nearest-Neighbour Representations of Boolean Functions
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中文总结 AI 辅助
本研究探讨布尔函数最近邻表示所需原型数，强化下界并给出VC维界，通过覆盖设计精确刻画对称阈值函数，并证明随机函数需约半数原型，对称函数需11/20。
中文摘要 AI 辅助
我们研究了通过最近邻分类表示布尔函数所需的原型数量。存在两种不同的设置：原型可以是欧几里得空间的任意点，或者原型本身必须位于布尔立方体中。对于无限制的原型，我们加强了几乎所有布尔函数的已知下界。该下界同时适用于具有任意数量投票邻居的最近邻投票规则，并显著缩小了与已知一般上界之间的差距。我们获得了具有有界原型数量的类的VC维界，并证明在至少四维空间中该界在阶上是尖锐的。然后我们研究布尔原型，从对称阈值函数开始。与覆盖设计的联系将每个阈值水平所需的最小原型数量精确地用覆盖数表示，并导致相关单调函数的进一步精确结果，包括析取扩展以及何时可以用单个负原型进行表示的特征描述。对于均匀随机的布尔函数，布尔最近邻复杂度作为立方体的比例，渐近地接近二分之一或一，并具有明确的极限概率。特别是，几乎每个布尔函数至少需要大约立方体中点数一半的原型，而二分之一是此类下界成立的最大比例。最后，我们考虑任意对称布尔函数。它们的布尔最近邻复杂度由路径上的加权顶点覆盖问题紧密近似。因此，均匀随机对称函数通常需要占立方体11/20的原型。这远大于当原型允许位于欧几里得空间任意位置时已知的上界。
英文摘要
We study the number of prototypes needed to represent Boolean functions by nearest-neighbour classification. There are two distinct settings: the prototypes may be arbitrary points of Euclidean space, or they may themselves be required to lie in the Boolean cube. For unrestricted prototypes, we strengthen a known lower bound for almost all Boolean functions. The bound applies simultaneously to nearest-neighbour voting rules with any number of voting neighbours, and substantially narrows the gap with the known general upper bound. We obtain a VC-dimension bound for classes with a bounded number of prototypes, and show that it is sharp in order in dimensions at least four. We then study Boolean prototypes, beginning with symmetric threshold functions. A connection with covering designs expresses the minimum number of prototypes at every threshold level exactly in terms of a covering number, and leads to further exact results for related monotone functions, including disjunctive extensions and a characterisation of when a representation with a single negative prototype is possible. For a uniformly random Boolean function, the Boolean nearest-neighbour complexity, as a proportion of the cube, is asymptotically close either to one half or to one, with explicit limiting probabilities. In particular, almost every Boolean function requires at least approximately half as many prototypes as there are points in the cube, and one half is the largest proportion for which such a lower bound holds. Finally, we consider arbitrary symmetric Boolean functions. Their Boolean nearest-neighbour complexity is closely approximated by a weighted vertex-cover problem on paths. As a consequence, a uniformly random symmetric function typically requires prototypes amounting to $11/20$ of the cube. This is much larger than the upper bounds known when the prototypes are allowed to lie anywhere in Euclidean space.
发表机构
- London School of Economics and Political Science（伦敦政治经济学院）
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