Outlier Detection Using Generative Models With Theoretical Performance Guarantees

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 by solving an <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\ell _{1}$ </tex-math></inline-formula> norm minimization problem. 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 and nonlinear generator neural networks with an arbitrary number of layers. We propose an iterative and linearized alternating direction method of multipliers (ADMM) algorithm for solving the outlier detection problem via <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\ell _{1}$ </tex-math></inline-formula> norm minimization, and a gradient descent algorithm for solving the outlier detection problem via squared <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\ell _{1}$ </tex-math></inline-formula> 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 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\ell _{2}$ </tex-math></inline-formula> norm minimization approach.