This paper proposes to learn Multi-task, Multi-modal Direct Acyclic Graphs (MM-DAGs), which are commonly observed in complex systems, e.g., traffic, manufacturing, and weather systems, whose variables are multi-modal with scalars, vectors, and functions. This paper takes the traffic congestion analysis as a concrete case, where a traffic intersection is usually regarded as a DAG. In a road network of multiple intersections, different intersections can only have some overlapping and distinct variables observed. For example, a signalized intersection has traffic light-related variables, whereas unsignalized ones do not. This encourages the multi-task design: with each DAG as a task, the MM-DAG tries to learn the multiple DAGs jointly so that their consensus and consistency are maximized. To this end, we innovatively propose a multi-modal regression for linear causal relationship description of different variables. Then we develop a novel Causality Difference (CD) measure and its differentiable approximator. Compared with existing SOTA measures, CD can penalize the causal structural difference among DAGs with distinct nodes and can better consider the uncertainty of causal orders. We rigidly prove our design's topological interpretation and consistency properties. We conduct thorough simulations and one case study to show the effectiveness of our MM-DAG. The code is available under https://github.com/Lantian72/MM-DAG

翻译：本文提出学习多任务、多模态有向无环图（MM-DAGs）。这类图结构在交通、制造及天气系统等复杂系统中普遍存在，其变量包含标量、向量及函数等多模态数据。本文以交通拥堵分析为具体案例，将交叉路口通常建模为有向无环图。在多交叉口的路网中，不同路口仅存在部分重叠及差异化的观测变量——例如，信号灯路口包含与交通灯相关的变量，而无信号灯路口则不具备此类变量。这一现象促使我们采用多任务设计：将每个有向无环图视为一个任务，MM-DAG尝试联合学习多个有向无环图，以最大化其共识性与一致性。为此，我们创新性地提出一种多模态回归方法，用于描述不同变量间的线性因果关系。进而开发了新的因果差异（CD）度量及其可微分近似器。与现有最优度量相比，CD能对具有不同节点的有向无环图间的因果结构差异进行惩罚，并更好地考虑因果顺序的不确定性。我们严格证明了所提设计的拓扑解释性与一致性性质。通过充分的仿真实验与一项案例研究，验证了MM-DAG的有效性。相关代码已开源在 https://github.com/Lantian72/MM-DAG