论文标题
分形维度，近似和数据集
Fractal dimension, approximation and data sets
论文作者
论文摘要
本文的目的是研究大型数据集中的分形现象以及降低维度的相关问题。我们研究了经典主体分析在识别数据集的显着分形特征无效的情况下。取而代之的是，我们采用了离散能量，这是一种从几何措施理论借来的技术，以限制位于$ k $维的超平面附近的给定数据集的数量，或者更一般而言，在一组给定的Minkowski尺寸附近。描述了由自然出现的数据集引起的具体动机，并概述了未来的方向。
The purpose of this paper is to study the fractal phenomena in large data sets and the associated questions of dimension reduction. We examine situations where the classical Principal Component Analysis is not effective in identifying the salient underlying fractal features of the data set. Instead, we employ the discrete energy, a technique borrowed from geometric measure theory, to limit the number of points of a given data set that lie near a $k$-dimensional hyperplane, or, more generally, near a set of a given upper Minkowski dimension. Concrete motivations stemming from naturally arising data sets are described and future directions outlined.
