Learning Max-Stable Representations that Extrapolate

High-dimensional images and measurements may indicate extreme and rare behavior, such as advanced stages of cancer or material failure, and understanding these distributions is important for many disciplines. Classically, extreme value theory (EVT) provides a theoretical framework for extrapolating based on extremeness of magnitude of measurements and is used to model the tail of a distribution from limited observations. We propose a framework for learning representations from high dimensional observations that are amenable to analysis using classical EVT. Specifically, we propose extending the $\max$-stability property of EVT to $\varphi$-stability, which generalizes the $\max$ operator to a more general operator $\varphi$ that has practical applications in high dimensional cases. We base $\varphi$-stability on representation learning techniques such that the resulting representations lend themselves to analysis by EVT and can model high-dimensional observations of extreme characteristics. This enables our method to extrapolate to observations of extreme behavior in the observation domain. We then extend our method to infinite dimensional observations such as time series. Empirical results indicate the utility of using max-stability for representations to extrapolate beyond the training data.