Unsupervised learning seeks to uncover patterns in data. However, different kinds of noise may impede the discovery of useful substructure from real-world time-series data. In this work, we focus on mitigating the interference of left-censorship in the task of clustering. We provide conditions under which clusters and left-censorship may be identified; motivated by this result, we develop a deep generative, continuous-time model of time-series data that clusters while correcting for censorship time. We demonstrate accurate, stable, and interpretable results on synthetic data that outperform several benchmarks. To showcase the utility of our framework on real-world problems, we study how left-censorship can adversely affect the task of disease phenotyping, resulting in the often incorrect assumption that longitudinal patient data are aligned by disease stage. In reality, patients at the time of diagnosis are at different stages of the disease -- both late and early due to differences in when patients seek medical care and such discrepancy can confound unsupervised learning algorithms. On two clinical datasets, our model corrects for this form of censorship and recovers known clinical subtypes.
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