Many data-science problems involve complex data evolving over time or space and governed by latent structures. Recovering these hidden representations is essential for characterizing dynamics and achieving generalization beyond observed data. This talk presents a unified framework for learning continuous latent structures under mixed membership, linking two domains: dynamic networks with irregular continuous-time interactions and nonparametric density mixtures where each observation arises from mixtures of latent generative densities. A central insight is that simplex geometry underlies these latent structures. We exploit this geometry using time-kernel smoothing for networks and topic-modeling–based unmixing for density mixtures. Leveraging random matrix theory and concentration inequalities, we develop methods that are statistically minimax-optimal, computationally efficient, and tuning-free. Applications include causal inference with interference on evolving networks and multi-agent reinforcement learning, where latent community structures support coordination and transfer.
