Multivariate information theory provides a general and principled framework for understanding how the components of a complex system are connected. Existing analyses are coarse in nature -- built up from characterizations of discrete subsystems -- and can be computationally prohibitive. In this work, we propose to study the continuous space of possible descriptions of a composite system as a window into its organizational structure. A description consists of specific information conveyed about each of the components, and the space of possible descriptions is equivalent to the space of lossy compression schemes of the components. We introduce a machine learning framework to optimize descriptions that extremize key information theoretic quantities used to characterize organization, such as total correlation and O-information. Through case studies on spin systems, Sudoku boards, and letter sequences from natural language, we identify extremal descriptions that reveal how system-wide variation emerges from individual components. By integrating machine learning into a fine-grained information theoretic analysis of composite random variables, our framework opens a new avenues for probing the structure of real-world complex systems.

Quantifying which neurons are important with respect to the classification decision of a trained neural network is essential for understanding their inner workings. Previous work primarily attributed importance to individual neurons. In this work, we study which groups of neurons contain synergistic or redundant information using a multivariate mutual information method called the O-information. We observe the first layer is dominated by redundancy suggesting general shared features (i.e. detecting edges) while the last layer is dominated by synergy indicating local class-specific features (i.e. concepts). Finally, we show the O-information can be used for multi-neuron importance. This can be demonstrated by re-training a synergistic sub-network, which results in a minimal change in performance. These results suggest our method can be used for pruning and unsupervised representation learning.