发表机构
Pontifícia Universidade Católica do Rio de Janeiro; Alfréd Rényi Institute of Mathematics; Loránd Eötvös University（里约热内卢天主教大学; 阿尔弗雷德·雷尼数学研究所; 罗兰·厄特沃什大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究实巴拿赫空间中扩展球等几何PCC的VC维数，通过映射线性化距离，利用平衡符号和估计等方法，得到\(L_p(\mu)\)中扩展球PCC的VC维数无维数上界及匹配下界，还推导了密集邻域引理。
AI 中文摘要
继Alon、Hanneke、Holzman和Moran（FOCS 2021）之后，我们将部分概念类（PCC）定义为一族部分函数\(f: V\to\{0,1,\ast\}\)；等价地，其概念将基础集划分为黑色（\(f^{-1}(1)\)）、灰色（\(f^{-1}(\ast)\)）和白色部分（\(f^{-1}(0)\)）。其VC维数由不取值\(\ast\)的破碎集定义。我们研究实巴拿赫空间中的两种几何PCC，都有边距\(\delta>0\)：扩展半空间，其中灰色部分是与半空间相邻的宽度至少为\(\delta\)的条带；扩展球，其中灰色部分是围绕单位半径球的宽度为\(\delta\)的环带。我们的主要结果是\(L_p(\mu)\)（\(1\le p<\infty\)）中扩展球的PCC的VC维数的无维数上界，包括非欧几里得且算法上特别相关的情况\(\ell^d_1\)。这些界取决于边距和半径，但不取决于环境维数或基础测度空间。这些是Bourneuf、Charbit和Thomassé（FOCS 2025）工作的扩展，他们研究了欧几里得空间即\(\ell_2^d\)中扩展球的PCC。我们还证明了VC维数的下界，在边距参数\(\delta\)方面与上界匹配。最后，我们在\(L_p\)空间中推导了一个密集邻域引理，同样扩展了已知的欧几里得结果。我们的方法依赖于通过映射到非平凡拉德马赫型空间来线性化距离，然后使用平衡符号和估计或无维数拉东定理。论证依赖于泛函分析的思想，并为该领域的非专家进行了清晰解释。
英文摘要
Following Alon, Hanneke, Holzman, and Moran (FOCS 2021), we define a partial concept class (PCC) as a family of partial functions \(f: V\to\{0,1,\ast\}\); equivalently, its concepts partition the ground set into black ($f^{-1}(1)$), grey ($f^{-1}(\ast)$), and white parts ($f^{-1}(0)$). Its VC dimension is defined by shattering sets on which the value $\ast$ is not taken. We study two geometric PCCs in real Banach spaces, both with a margin \(δ>0\): expanded half-spaces, where the grey part is a strip of width at least \(δ\) adjacent to a half-space, and expanded balls, where the grey part is an annulus of width \(δ\) around a unit radius ball. Our main results are dimension-free upper bounds on the VC dimension of the PCC of expanded balls in \(L_p\parenthμ\), \(1\le p<\infty\), including the non-Euclidean and algorithmically particularly relevant case \(\ell^d_1\). These bounds depend on the margin and on the radii, but not on the ambient dimension or the underlying measure space. These are extensions of the work of Bourneuf, Charbit, and Thomassé (FOCS 2025) who studied the PCC of expanded balls in Euclidean space, that is, $\ell_2^d$. We also prove lower bounds on the VC dimension that match the upper bounds in terms of the margin parameter $δ$. Finally, we derive a Dense Neighborhood Lemma in \(L_p\)-spaces, again extending the known Euclidean results. Our method relies on the linearization of the distance through a map into a space of non-trivial Rademacher type, and then the use of a balanced signed-sum estimate, or a no-dimensional Radon theorem. The arguments rely on ideas from functional analysis that are clearly explained for the non-expert in that field.
Comments22 pages
