论文标题
主椭圆形分析（PEA）：有效的非线性尺寸降低和聚类
Principal Ellipsoid Analysis (PEA): Efficient non-linear dimension reduction & clustering
论文作者
论文摘要
即使过度参数化模型的普及程度升高，简单的维度降低和聚类方法（例如PCA和K-均值）仍在各种环境中通常使用。主要原因是简单性，可解释性和计算效率的结合。本文的重点是通过允许数据中的非线性关系和更灵活的群集形状，而无需牺牲关键优势来改善PCA和K-均值。关键的贡献是用于主椭圆分析（PEA）的新框架，它定义了一种简单且在计算上有效的PCA替代方案，该替代品可以通过数据符合最佳的椭圆形近似。我们使用VAPNIK-CHERVONENKIS（VC）理论提供了理论保证，以表现出强烈的一致性和均匀的浓度界限。玩具实验说明了PEA的性能以及适应非线性结构和复杂簇形状的能力。在各种真实的数据聚类应用程序中，PEA与简单数据集的K均值一样，同时显着改善了更复杂的设置的性能。
Even with the rise in popularity of over-parameterized models, simple dimensionality reduction and clustering methods, such as PCA and k-means, are still routinely used in an amazing variety of settings. A primary reason is the combination of simplicity, interpretability and computational efficiency. The focus of this article is on improving upon PCA and k-means, by allowing non-linear relations in the data and more flexible cluster shapes, without sacrificing the key advantages. The key contribution is a new framework for Principal Elliptical Analysis (PEA), defining a simple and computationally efficient alternative to PCA that fits the best elliptical approximation through the data. We provide theoretical guarantees on the proposed PEA algorithm using Vapnik-Chervonenkis (VC) theory to show strong consistency and uniform concentration bounds. Toy experiments illustrate the performance of PEA, and the ability to adapt to non-linear structure and complex cluster shapes. In a rich variety of real data clustering applications, PEA is shown to do as well as k-means for simple datasets, while dramatically improving performance in more complex settings.
