Defining a successful notion of a multivariate quantile has been an open problem for more than half a century, motivating a plethora of possible solutions. Of these, the approach of [8] and [25] leading to M-quantiles, is very appealing for its mathematical elegance combining elements of convex analysis and probability theory. The key idea is the description of a convex function (the K-function) whose gradient (the K-transform) is in one-to-one correspondence between all of R^d and the unit ball in R^d. By analogy with the d=1 case where the K-transform is a cumulative distribution function-like object (an M-distribution), the fact that its inverse is guaranteed to exist lends itself naturally to providing the basis for the definition of a quantile function for all d>=1. Over the past twenty years the resulting M-quantiles have seen applications in a variety of fields, primarily for the purpose of detecting outliers in multidimensional spaces. In this article we prove that for odd d>=3, it is not the gradient but a poly-Laplacian of the K-function that is (almost everywhere) proportional to the density function. For d even one cannot establish a differential equation connecting the K-function with the density. These results show that usage of the K-transform for outlier detection in higher odd-dimensions is in principle flawed, as the K-transform does not originate from inversion of a true M-distribution. We demonstrate these conclusions in two dimensions through examples from non-standard asymmetric distributions. Our examples illustrate a feature of the K-transform whereby regions in the domain with higher density map to larger volumes in the co-domain, thereby producing a magnification effect that moves inliers closer to the boundary of the co-domain than outliers. This feature obviously disrupts any outlier detection mechanism that relies on the inverse K-transform.

翻译：定义多变量分位数的成功概念半个多世纪以来一直是一个开放问题，催生了大量可能的解决方案。其中，[8]和[25]提出的基于M-分位数的方法因其结合凸分析与概率论要素的数学优雅性而极具吸引力。其核心思想是描述一个凸函数（K-函数），其梯度（K-变换）在R^d空间与R^d单位球之间建立一一对应关系。类比d=1情形（此时K-变换类似于累积分布函数对象即M-分布），其逆映射必然存在的事实自然为所有d≥1情形下分位数函数的定义提供了基础。过去二十年中，由此产生的M-分位数已在多个领域得到应用，主要用於检测多维空间中的异常值。本文证明，对于奇数d≥3情形，与密度函数（几乎处处）成比例的并非K-函数的梯度，而是其多阶拉普拉斯算子。对于偶数d情形，无法建立连接K-函数与密度函数的微分方程。这些结果表明，在更高奇数维空间中使用K-变换进行异常检测在本质上存在缺陷，因为K-变换并非源自真实M-分布的逆变换。我们通过非标准非对称分布实例在二维空间中展示了这些结论。我们的实例揭示了K-变换的一个特征：域中密度较高的区域会映射到共域中更大的体积，从而产生放大效应——将内部点推至比异常点更靠近共域边界的位置。这一特征显然破坏了任何依赖逆K-变换的异常检测机制。