论文标题
使用基于RKHS的不确定性力矩和最佳运输的稳健依赖度量
Robust Dependence Measure using RKHS based Uncertainty Moments and Optimal Transport
论文作者
论文摘要
在统计和机器学习的许多应用中，变量之间的依赖性可靠测量至关重要。当前的依赖估计方法，尤其是基于密度的方法，缺乏精度，鲁棒性和/或解释性（就估计的依赖类型而言）。我们提出了一种随机变量之间的依赖性定量的两步方法：1）我们首先分解了在多个不确定性的局部不确定性方面所涉及的变量的概率密度函数（PDF），这些变量是系统地，精确地识别PDF的不同区域（在尾部区域特别强调）。 2）然后，我们计算一个最佳传输图，以测量变量分解的局部不确定性矩之间的几何相似性。然后，依赖性由最佳传输图中变量的各个不确定性矩之间的一对一对应度确定。我们利用了最近引入的高斯繁殖Hilbert Space（RKHS）的框架，用于变量的多态不确定性分解。基于高斯RKHS，我们的方法对数据的异常值和单调转换是强大的，而不确定性的多个矩可提供量化依赖类型的高分辨率和可解释性。我们使用模拟数据通过一些初步结果来支持这些主张。
Reliable measurement of dependence between variables is essential in many applications of statistics and machine learning. Current approaches for dependence estimation, especially density-based approaches, lack in precision, robustness and/or interpretability (in terms of the type of dependence being estimated). We propose a two-step approach for dependence quantification between random variables: 1) We first decompose the probability density functions (PDF) of the variables involved in terms of multiple local moments of uncertainty that systematically and precisely identify the different regions of the PDF (with special emphasis on the tail-regions). 2) We then compute an optimal transport map to measure the geometric similarity between the corresponding sets of decomposed local uncertainty moments of the variables. Dependence is then determined by the degree of one-to-one correspondence between the respective uncertainty moments of the variables in the optimal transport map. We utilize a recently introduced Gaussian reproducing kernel Hilbert space (RKHS) based framework for multi-moment uncertainty decomposition of the variables. Being based on the Gaussian RKHS, our approach is robust towards outliers and monotone transformations of data, while the multiple moments of uncertainty provide high resolution and interpretability of the type of dependence being quantified. We support these claims through some preliminary results using simulated data.
