  In this work, we propose a novel generative model for mapping inputs to
structured, high-dimensional outputs using structured conditional normalizing
flows and Gaussian process regression. The model is motivated by the need to
characterize uncertainty in the input/output relationship when making
inferences on new data. In particular, in the physical sciences, limited
training data may not adequately characterize future observed data; it is
critical that models adequately indicate uncertainty, particularly when they
may be asked to extrapolate. In our proposed model, structured conditional
normalizing flows provide parsimonious latent representations that relate to
the inputs through a Gaussian process, providing exact likelihood calculations
and uncertainty that naturally increases away from the training data inputs. We
demonstrate the methodology on laser-induced breakdown spectroscopy data from
the ChemCam instrument onboard the Mars rover Curiosity. ChemCam was designed
to recover the chemical composition of rock and soil samples by measuring the
spectral properties of plasma atomic emissions induced by a laser pulse. We
show that our model can generate realistic spectra conditional on a given
chemical composition and that we can use the model to perform uncertainty
quantification of chemical compositions for new observed spectra. Based on our
results, we anticipate that our proposed modeling approach may be useful in
other scientific domains with high-dimensional, complex structure where it is
important to quantify predictive uncertainty.
