  Learning data representations under uncertainty is an important task that
emerges in numerous machine learning applications. However, uncertainty
quantification (UQ) techniques are computationally intensive and become
prohibitively expensive for high-dimensional data. In this paper, we present a
novel surrogate model for representation learning and uncertainty
quantification, which aims to deal with data of moderate to high dimensions.
The proposed model combines a neural network approach for dimensionality
reduction of the (potentially high-dimensional) data, with a surrogate model
method for learning the data distribution. We first employ a variational
autoencoder (VAE) to learn a low-dimensional representation of the data
distribution. We then propose to harness polynomial chaos expansion (PCE)
formulation to map this distribution to the output target. The coefficients of
PCE are learned from the distribution representation of the training data using
a maximum mean discrepancy (MMD) approach. Our model enables us to (a) learn a
representation of the data, (b) estimate uncertainty in the high-dimensional
data system, and (c) match high order moments of the output distribution;
without any prior statistical assumptions on the data. Numerical experimental
results are presented to illustrate the performance of the proposed method.
