We propose a new and intuitive metric for aleatoric uncertainty quantification (UQ), the prevalence of class collisions defined as the same input being observed in different classes. We use the rate of class collisions to define the collision matrix, a novel and uniquely fine-grained measure of uncertainty. For a classification problem involving $K$ classes, the $K\times K$ collision matrix $S$ measures the inherent difficulty in distinguishing between each pair of classes. We discuss several applications of the collision matrix, establish its fundamental mathematical properties, and show its relationship with existing UQ methods, including the Bayes error rate (BER). We also address the new problem of estimating the collision matrix using one-hot labeled data by proposing a series of innovative techniques to estimate $S$. First, we learn a pair-wise contrastive model which accepts two inputs and determines if they belong to the same class. We then show that this contrastive model (which is PAC learnable) can be used to estimate the row Gramian matrix of $S$, defined as $G=SS^T$. Finally, we show that under reasonable assumptions, $G$ can be used to uniquely recover $S$, a new result on non-negative matrices which could be of independent interest. With a method to estimate $S$ established, we demonstrate how this estimate of $S$, in conjunction with the contrastive model, can be used to estimate the posterior class probability distribution of any point. Experimental results are also presented to validate our methods of estimating the collision matrix and class posterior distributions on several datasets.

翻译：本文提出了一种新颖且直观的偶然不确定性量化（UQ）度量标准，即类别碰撞的普遍性，其定义为同一输入在不同类别中被观测到的现象。我们利用类别碰撞率定义了碰撞矩阵，这是一种新颖且具有独特细粒度特性的不确定性度量方法。对于一个涉及 $K$ 个类别的分类问题，$K\times K$ 的碰撞矩阵 $S$ 衡量了区分每一对类别之间的固有难度。我们讨论了碰撞矩阵的若干应用，建立了其基本数学性质，并展示了其与现有UQ方法（包括贝叶斯错误率（BER））的关系。我们还探讨了使用独热编码标注数据估计碰撞矩阵这一新问题，并提出了一系列创新技术来估计 $S$。首先，我们学习一个成对对比模型，该模型接受两个输入并判断它们是否属于同一类别。然后，我们证明这个（可PAC学习的）对比模型可用于估计 $S$ 的行格拉姆矩阵，定义为 $G=SS^T$。最后，我们证明在合理的假设下，$G$ 可用于唯一地恢复 $S$，这是关于非负矩阵的一个新结果，可能具有独立的研究价值。在建立了估计 $S$ 的方法后，我们演示了如何利用 $S$ 的估计值，结合对比模型，来估计任意点的后验类别概率分布。我们还提供了在多个数据集上的实验结果，以验证我们估计碰撞矩阵和类别后验分布方法的有效性。