  This paper considers the problem of recovering signals modeled by generative
models from linear measurements contaminated with sparse outliers. We propose
an outlier detection approach for reconstructing the ground-truth signals
modeled by generative models under sparse outliers. We establish theoretical
recovery guarantees for reconstruction of signals using generative models in
the presence of outliers, giving lower bounds on the number of correctable
outliers. Our results are applicable to both linear generator neural networks
and the nonlinear generator neural networks with an arbitrary number of layers.
We propose an iterative alternating direction method of multipliers (ADMM)
algorithm for solving the outlier detection problem via $\ell_1$ norm
minimization, and a gradient descent algorithm for solving the outlier
detection problem via squared $\ell_1$ norm minimization. We conduct extensive
experiments using variational auto-encoder and deep convolutional generative
adversarial networks, and the experimental results show that the signals can be
successfully reconstructed under outliers using our approach. Our approach
outperforms the traditional Lasso and $\ell_2$ minimization approach.
