主多项式分析
Principal Polynomial Analysis
摘要：
本文提出了一种新的流形学习框架，基于一系列的主要多项式，捕捉可能的非线性性质的数据。建议的主多项式分析（PPA）推广PCA建模的最大方差的方向，通过曲线，而不是直线。与以前的方法不同，PPA减少到执行简单的单变量回归，这使得它在计算上是可行的和鲁棒的。此外，PPA显示了一些有趣的分析性质。首先，PPA是一个保体积映射，这反过来保证了逆的存在性。第二，这样的逆可以以封闭形式获得。与其他学习方法相比，可逆性是一个重要的优势，因为它允许理解数据具有物理意义的输入域中的识别特征。此外，它允许在合理的（输入域）单元中评估降维的性能。体积保持还允许信息理论量的容易计算，诸如变换之后的多信息的减少。第三，PPA的分析性质导致了流形的清晰几何解释：它允许计算Frenet-Serret框架（局部特征）和空间任何点的广义曲率。第四，分析雅可比矩阵允许计算由数据引起的度量，从而推广了马氏距离。这些特性的理论和实验证明。PPA的性能进行了评估，在维数和冗余减少，在合成和真实的数据集从UCI存储库。
英文摘要：
This paper presents a new framework for manifold learning based on a sequence of principal polynomials that capture the possibly nonlinear nature of the data. The proposed Principal Polynomial Analysis (PPA) generalizes PCA by modeling the directions of maximal variance by means of curves, instead of straight lines. Contrarily to previous approaches, PPA reduces to performing simple univariate regressions, which makes it computationally feasible and robust. Moreover, PPA shows a number of interesting analytical properties. First, PPA is a volume-preserving map, which in turn guarantees the existence of the inverse. Second, such an inverse can be obtained in closed form. Invertibility is an important advantage over other learning methods, because it permits to understand the identified features in the input domain where the data has physical meaning. Moreover, it allows to evaluate the performance of dimensionality reduction in sensible (input-domain) units. Volume preservation also allows an easy computation of information theoretic quantities, such as the reduction in multi-information after the transform. Third, the analytical nature of PPA leads to a clear geometrical interpretation of the manifold: it allows the computation of Frenet-Serret frames (local features) and of generalized curvatures at any point of the space. And fourth, the analytical Jacobian allows the computation of the metric induced by the data, thus generalizing the Mahalanobis distance. These properties are demonstrated theoretically and illustrated experimentally. The performance of PPA is evaluated in dimensionality and redundancy reduction, in both synthetic and real datasets from the UCI repository.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
