通过碰撞进行细粒度不确定性量化
Fine-Grained Uncertainty Quantification via Collisions
AI总结：
本文提出了一种新的度量方法，通过类碰撞率来量化偶然不确定性，定义了碰撞矩阵作为新的细粒度不确定性度量，并展示了其与现有不确定性量化方法的关系。
摘要：
我们提出了一种新的直观度量方法，用于偶然不确定性量化（UQ），即相同输入被观测到不同类别的频率。我们使用类碰撞率来定义碰撞矩阵，这是一种新的、唯一细粒度的不确定性度量。对于涉及K类的分类问题，K×K的碰撞矩阵S衡量每对类之间固有的区分难度。我们讨论了碰撞矩阵的几种应用，建立了其基本数学性质，并展示了其与现有UQ方法的关系，包括贝叶斯错误率（BER）。我们还解决了使用one-hot标记数据估计碰撞矩阵的新问题，提出了一系列创新技术来估计S。首先，我们学习了一个成对对比模型，接受两个输入并确定它们是否属于同一类。然后，我们展示该对比模型（可被PAC学习）可用于估计S的行Gramian矩阵，定义为G=SS^T。最后，我们展示在合理假设下，G可用于唯一恢复S，这在非负矩阵方面可能有独立兴趣。通过建立估计S的方法，我们展示了如何利用S的估计与对比模型来估计任意点的后验类概率分布。实验结果也展示了在多个数据集上我们估计碰撞矩阵和类后验分布方法的验证。
英文摘要：
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.
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