In the field of information fusion, the problem of data aggregation has been formalized as an order-preserving process that builds upon the property of monotonicity. However, fields such as computational statistics, data analysis, and geometry usually emphasize the role of equivariances to various geometrical transformations in aggregation processes. Admittedly, if we consider a unidimensional data fusion task, both requirements are often compatible with each other. Nevertheless, in this paper, we show that, in the multidimensional setting, the only idempotent functions that are monotone and orthogonal equivariant are the over-simplistic weighted centroids. Even more, this result still holds after replacing monotonicity and orthogonal equivariance by the weaker property of orthomonotonicity. This implies that the aforementioned approaches to the aggregation of multidimensional data are irreconcilable, and that, if a weighted centroid is to be avoided, we must choose between monotonicity and a desirable behavior with regard to orthogonal transformations.

中文翻译：

---

多维数据聚合的固有困难


在信息融合领域，数据聚合问题已被形式化为基于单调性的保序过程。然而，计算统计、数据分析和几何等领域通常强调等方差在聚合过程中对各种几何变换的作用。诚然，如果我们考虑一维数据融合任务，这两个要求通常是相互兼容的。然而，在本文中，我们表明，在多维环境中，唯一单调且正交等变的幂等函数是过于简单化的加权质心。更重要的是，用较弱的正交单调性代替单调性和正交等变性后，这个结果仍然成立。这意味着上述多维数据聚合的方法是不可调和的，并且如果要避免加权质心，我们必须在单调性和正交变换方面的理想行为之间进行选择。