Title: A Mathematical Framework for Characterizing Dependency Structures of Multimodal Learning. Abstract: Dependency structures between modalities have been utilized explicitly and implicitly in multimodal learning to enhance classification performance, particularly when the training samples are insufficient. Recent efforts have concentrated on developing suitable dependence structures and applying them in deep neural networks, but the interplay between the training sample size and various structures has not received enough attention. To address this issue, we propose a mathematical framework that can be utilized to characterize conditional dependency structures in analytic ways. It provides an explicit description of the sample size in learning various structures in a non-asymptotic regime. Additionally, it demonstrates how task complexity and a fitness evaluation of conditional dependence structures affect the results. Furthermore, we develop an autonomous updated coefficient algorithm auto-CODES based on the theoretical framework and conduct experiments on multimodal emotion recognition tasks using the MELD and IEMOCAP datasets. The experimental results validate our theory and show the effectiveness of the proposed algorithm.