基于正半定散度矩阵的不变坐标选择的广义实现
Generalized implementation of invariant coordinate selection with positive semi-definite scatter matrices
AI总结：
本文针对不变坐标选择中散度矩阵奇异导致经典方法不可解的问题，提出基于Moore-Penrose伪逆、降维和广义奇异值分解的三种扩展，并验证其在异常检测、聚类及高维低样本量数据中的适用性。
摘要：
不变坐标选择是一种无监督的多元数据变换方法，在异常值检测或聚类等许多场景中非常有用。它基于两个仿射等变且正定的散度矩阵的同时对角化。其经典实现依赖于一个非对称特征值问题，即相对于其中一个散度矩阵对另一个散度矩阵进行对角化。在存在共线性的情况下，至少有一个散度矩阵是奇异的，这使得问题无法求解。为了解决这一局限性，本文提出了三种方法，分别使用：Moore-Penrose 伪逆、降维以及广义奇异值分解。我们从理论上并通过各种实证应用研究了这些方法的性质。总体而言，基于广义奇异值分解的扩展似乎最有前景，尽管它将散度矩阵的选择限制为那些可以表示为交叉乘积的矩阵。在实践中，其中一些方法也适用于高维低样本量数据的情形。
英文摘要：
Invariant coordinate selection is an unsupervised multivariate data transformation useful in many contexts such as outlier detection or clustering. It is based on the simultaneous diagonalization of two affine equivariant and positive definite scatter matrices. Its classical implementation relies on a non-symmetric eigenvalue problem by diagonalizing one scatter relatively to the other. In case of collinearity, at least one of the scatter matrices is singular, making the problem unsolvable. To address this limitation, three approaches are proposed using: a Moore-Penrose pseudo inverse, a dimension reduction, and a generalized singular value decomposition. Their properties are investigated both theoretically and through various empirical applications. Overall, the extension based on the generalized singular value decomposition seems the most promising, even though it restricts the choice of scatter matrices to those that can be expressed as cross-products. In practice, some of the approaches also appear suitable in the context of data in high-dimension low-sample-size data.
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