Uncovering high-order patterns of data with spin models: when simplicity matters

Abstract

Finding the model that best captures patterns hidden within noisy data is a central problem in science. In this context, the Ising model has been widely used to infer pairwise patterns in binary data. In recent years, attention has been brought to high-order patterns of data and the question of how to detect them. We will discuss the use of (classical) spin models with interactions of order higher than two to extract such patterns in binary data, and why this problem is challenging. By analyzing the information-theoretic complexity of spin models, we will see that, despite their appearance, models with high-order interactions are not necessarily more complex than pairwise models.

We will then focus on a sub-family of spin models with minimal information-theoretic complexity, which we call Minimally Complex Models (MCMs). These models have interactions organized in a community-like manner, and we will see how this can be used to identify groups of highly correlated variables in binary data. Uncovering such community structures in noisy data is crucial to understanding emergent phenomena in many complex systems, such as the brain, or health or social systems. So far, existing techniques rely solely on the pairwise correlation patterns of the data; in contrast, our approach takes into account all higher-order data patterns. We will demonstrate the capabilities of our approach against pairwise community detection on artificial data with built-in high-order community structures, and discuss the consequence of using a pairwise approach when the data is inherently high-order. Finally, we will discuss possible applications of MCMs in the context of neuroscience. Using Minimally Complex Models opens up new ways to tackle high-dimensional data modeling.