This letter introduces a new framework for quantifying predictive uncertainty for both data and models that rely on projecting the data into a gaussian reproducing kernel Hilbert space (RKHS) and transforming the data probability density function (PDF) in a way that quantifies the flow of its gradient as a topological potential field quantified at all points in the sample space. This enables the decomposition of the PDF gradient flow by formulating it as a moment decomposition problem using operators from quantum physics, specifically Schrödinger's formulation. We experimentally show that the higher-order modes systematically cluster the different tail regions of the PDF, thereby providing unprecedented discriminative resolution of data regions having high epistemic uncertainty. In essence, this approach decomposes local realizations of the data PDF in terms of uncertainty moments. We apply this framework as a surrogate tool for predictive uncertainty quantification of point-prediction neural network models, overcoming various limitations of conventional Bayesian-based uncertainty quantification methods. Experimental comparisons with some established methods illustrate performance advantages that our framework exhibits.

这封信引入了一个新框架，用于量化数据和模型的预测不确定性，该框架依赖于将数据投影到高斯再现核希尔伯特空间 (RKHS) 并以量化其流动的方式转换数据概率密度函数 (PDF)。梯度作为在样本空间中的所有点量化的拓扑势场。这可以通过使用量子物理学中的算子（特别是薛定谔公式）将其表述为矩分解问题来分解 PDF 梯度流。我们通过实验表明，高阶模式系统地聚类了 PDF 的不同尾部区域，从而为具有高认知不确定性的数据区域提供了前所未有的判别分辨率。在本质上，这种方法根据不确定性矩分解了数据 PDF 的局部实现。我们将此框架用作点预测神经网络模型的预测不确定性量化的替代工具，克服了传统的基于贝叶斯的不确定性量化方法的各种限制。与一些既定方法的实验比较说明了我们的框架所展示的性能优势。