  Many experimental time series measurements share unobserved causal drivers.
Examples include genes targeted by transcription factors, ocean flows
influenced by large-scale atmospheric currents, and motor circuits steered by
descending neurons. Reliably inferring this unseen driving force is necessary
to understand the intermittent nature of top-down control schemes in diverse
biological and engineered systems. Here, we introduce a new unsupervised
learning algorithm that uses recurrences in time series measurements to
gradually reconstruct an unobserved driving signal. Drawing on the mathematical
theory of skew-product dynamical systems, we identify recurrence events shared
across response time series, which implicitly define a recurrence graph with
glass-like structure. As the amount or quality of observed data improves, this
recurrence graph undergoes a percolation transition manifesting as weak
ergodicity breaking for random walks on the induced landscape -- revealing the
shared driver's dynamics, even in the presence of strongly corrupted or noisy
measurements. Across several thousand random dynamical systems, we empirically
quantify the dependence of reconstruction accuracy on the rate of information
transfer from a chaotic driver to the response systems, and we find that
effective reconstruction proceeds through gradual approximation of the driver's
dominant orbit topology. Through extensive benchmarks against classical and
neural-network-based signal processing techniques, we demonstrate our method's
strong ability to extract causal driving signals from diverse real-world
datasets spanning ecology, genomics, fluid dynamics, and physiology.
