Foundation models like chatGPT have demonstrated remarkable performance on various tasks. However, for many questions, they may produce false answers that look accurate. How do we train the model to precisely understand the concepts? In this paper, we introduce succinct representations of concepts based on category theory. Such representation yields concept-wise invariance properties under various tasks, resulting a new learning algorithm that can provably and accurately learn complex concepts or fix misconceptions. Moreover, by recursively expanding the succinct representations, one can generate a hierarchical decomposition, and manually verify the concept by individually examining each part inside the decomposition.

翻译：像chatGPT这样的基础模型在各种任务中展现了卓越的性能。然而，对于许多问题，它们可能产生看似准确但实际错误的答案。我们如何训练模型精确理解概念？本文基于范畴论引入了概念的简洁表示。这种表示在各种任务下产生了概念层面的不变性属性，由此得到一种新的学习算法，能够可证明地准确学习复杂概念或纠正错误认知。此外，通过递归扩展简洁表示，可以生成层次化分解，并通过逐一检查分解中的每个部分来人工验证概念。