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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 withscalars, vectors, and functions. This paper takes the traffic congestionanalysis as a concrete case, where a traffic intersection is usually regardedas a DAG. In a road network of multiple intersections, different intersectionscan only have some overlapping and distinct variables observed. For example, asignalized intersection has traffic light-related variables, whereasunsignalized ones do not. This encourages the multi-task design: with each DAGas a task, the MM-DAG tries to learn the multiple DAGs jointly so that theirconsensus and consistency are maximized. To this end, we innovatively propose amulti-modal regression for linear causal relationship description of differentvariables. Then we develop a novel Causality Difference (CD) measure and itsdifferentiable approximator. Compared with existing SOTA measures, CD canpenalize the causal structural difference among DAGs with distinct nodes andcan better consider the uncertainty of causal orders. We rigidly prove ourdesign's topological interpretation and consistency properties. We conductthorough simulations and one case study to show the effectiveness of ourMM-DAG. The code is available under https://github.com/Lantian72/MM-DAG

MM-DAG: Multi-task DAG Learning for Multi-modal Data -- with Application for Traffic Congestion Analysis