A Factor Model Based Time-Varying Auto-Align Overlapping Clustering

In many scientific fields such as genomics and neuroscience, data often exhibit complex structures where features can belong to multiple clusters simultaneously and vary over time. Traditional clustering methods such as k-means and hierarchical clustering typically assume static cluster structures and perform hard assignments, limiting their ability to capture the underlying overlapping and time-varying cluster structures. To address these challenges, we propose a novel time-varying Auto-Align clustering (AAclust) framework that effectively estimates and tracks dynamic changes in overlapping cluster memberships over time. Specifically, we employ a time-varying factor model, in which the overlapping cluster structures are modeled through a factor loading matrix that is allowed to vary over time. Theoretically, we show that the population factor loading matrix is identifiable up to a signed permutation across time, i.e., the population overlapping cluster memberships are identifiable. Furthermore, we show that our framework can consistently estimate the number of clusters and the overlapping cluster memberships for the features. By leveraging cross-covariance matrices between any two time points, our proposed method automatically aligns cluster labels over time without requiring post-hoc label matching based on similarity measures like the Rand index. We evaluate the performance of our proposed method through numerical experiments and an application to a functional magnetic resonance imaging dataset under continuous natural stimuli.