论文标题
n维的形状比例和球形
Shape Proportions and Sphericity in n Dimensions
论文作者
论文摘要
高尺寸的物体的形状指标仍然很稀疏。确实存在的那些（例如超体积）仍然仅限于更好地理解的物体，例如柏拉图固体和$ n $ ipubes。此外，了解较高维度的不确定形状的对象充其量是模棱两可的。过去的工作没有提供单个数字来给予对象的定性理解。例如，主要组件分析的特征值会导致$ n $指标来描述对象的形状。因此，我们需要一个数字，可以区分彼此不同形状的对象。先前的工作为特定维度（例如两个或三个维度）开发了形状指标。但是，有机会为任何所需的维度开发指标。为此，我们在给定数量的维度中为对象提供了两个新的形状指标：超球性和超形比例（SP）。我们以多种不同形状在内的$ n $ balls来探讨这些指标的所有权。然后，我们将这些指标连接到分析多维数据（例如流行IRIS数据集）形状的应用。
Shape metrics for objects in high dimensions remain sparse. Those that do exist, such as hyper-volume, remain limited to objects that are better understood such as Platonic solids and $n$-Cubes. Further, understanding objects of ill-defined shapes in higher dimensions is ambiguous at best. Past work does not provide a single number to give a qualitative understanding of an object. For example, the eigenvalues from principal component analysis results in $n$ metrics to describe the shape of an object. Therefore, we need a single number which can discriminate objects with different shape from one another. Previous work has developed shape metrics for specific dimensions such as two or three dimensions. However, there is an opportunity to develop metrics for any desired dimension. To that end, we present two new shape metrics for objects in a given number of dimensions: hyper-Sphericity and hyper-Shape Proportion (SP). We explore the proprieties of these metrics on a number of different shapes including $n$-balls. We then connect these metrics to applications of analyzing the shape of multidimensional data such as the popular Iris dataset.
