In domains such as ecological systems, collaborations, and the human brain the variables interact in complex ways. Yet accurately characterizing higher-order variable interactions (HOIs) is a difficult problem that is further exacerbated when the HOIs change across the data. To solve this problem we propose a new method called Local Correlation Explanation (CorEx) to capture HOIs at a local scale by first clustering data points based on their proximity on the data manifold. We then use a multivariate version of the mutual information called the total correlation, to construct a latent factor representation of the data within each cluster to learn the local HOIs. We use Local CorEx to explore HOIs in synthetic and real world data to extract hidden insights about the data structure. Lastly, we demonstrate Local CorEx's suitability to explore and interpret the inner workings of trained neural networks.