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，文献[8]和[25]提出的M-分位数方法因融合凸分析与概率论元素的数学优雅性而极具吸引力。其核心思想在于描述一个凸函数（K-函数），其梯度（K-变换）在R^d空间与R^d单位球面之间建立一一对应关系。通过与d=1情形（此时K-变换类似于累积分布函数对象，即M-分布）类比，其逆变换必然存在的特性自然为定义所有d≥1情形下的分位数函数提供了基础。过去二十年间，由此产生的M-分位数已在多个领域得到应用，主要用于多维空间中的异常点检测。本文证明：对于奇数d≥3情形，与密度函数成比例（几乎处处）的并非K-函数的梯度，而是其多Laplace算子；对于偶数d，则无法建立连接K-函数与密度函数的微分方程。这些结果表明，在奇数高维空间中使用K-变换进行异常检测存在根本性缺陷——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的二阶实例验证这些结论。我们的实例揭示了K-变换的特征：定义域中高密度区域映射至陪域中的更大体积，从而产生将内点推向陪域边界（相较于异常点）的放大效应。这一特性显然会破坏任何依赖逆K-变换的异常检测机制。