为低维数据分类提供数学框架，揭示数据结构对模型能力的影响。
Function-Counting Theory for Low-Dimensional Data Structures
- 基于柯弗理论改进假设，考虑数据低维特性
- 推导反映数据结构的分类二分计数方法
- 适用于研究数据结构如何影响模型性能
深度学习在分类与回归任务中的成功，常归因于真实世界数据虽高维表征却具有低维结构。本文旨在为低维数据上的二分类问题建立数学框架，基于柯弗（1965）的函数计数理论。原理论依赖一般位置假设，忽略数据内在结构。本文修正该假设以体现数据的低维性，推导出反映数据结构的分类二分计数。进一步将柯弗的分离能力与泛化问题拓展至低维设定，使数据结构对这两方面的影响得以分析。
原文摘要 · Abstract (English)
The success of deep learning models in classification and regression is widely attributed to the low-dimensional structure that real-world data tend to exhibit, despite their high-dimensional representation. This work attempts to provide a mathematical framework for binary classification on low-dimensional data, building on Cover's (1965) function-counting theory. With our framework, we aim to address the question of how the low-dimensional structure of the data affects the classification capabilities of learning models. Cover's theory relies on a general position assumption that blinds it to the underlying data structure. We refine this assumption to account for the low-dimensionality of the data and derive dichotomy counts that reflect the data structure. We further extend Cover's separation capacity and problem of generalization to the low-dimensional setting, enabling the impact of the underlying data structure on both to be analyzed.
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