In this task, you are given the abstract of a research paper. Your task is to generate a summary of this abstract. Your summary should not be very short, but it's better if it's not more than 30 words.

Input: Consider Input: Deep neural networks are known to be vulnerable to adversarial perturbations. In this paper, we bridge adversarial robustness of neural nets with Lyapunov stability of dynamical systems. From this viewpoint, training neural nets is equivalent to finding an optimal control of the discrete dynamical system, which allows one to utilize methods of successive approximations, an optimal control algorithm based on Pontryagin's maximum principle, to train neural nets. This decoupled training method allows us to add constraints to the optimization, which makes the deep model more robust. The constrained optimization problem can be formulated as a semi-definite programming problem and hence can be solved efficiently. Experiments show that our method effectively improves deep model's adversarial robustness.

Output: An adversarial defense method bridging robustness of deep neural nets with Lyapunov stability

Input: Consider Input: We propose a new learning-based approach to solve ill-posed inverse problems in imaging. We address the case where ground truth training samples are rare and the problem is severely ill-posed---both because of the underlying physics and because we can only get few measurements. This setting is common in geophysical imaging and remote sensing. We show that in this case the common approach to directly learn the mapping from the measured data to the reconstruction becomes unstable. Instead, we propose to first learn an ensemble of simpler mappings from the data to projections of the unknown image into random piecewise-constant subspaces. We then combine the projections to form a final reconstruction by solving a deconvolution-like problem. We show experimentally that the proposed method is more robust to measurement noise and corruptions not seen during training than a directly learned inverse.

Output: We solve ill-posed inverse problems with scarce ground truth examples by estimating an ensemble of random projections of the model instead of the model itself.

Input: Consider Input: We evaluate the distribution learning capabilities of generative adversarial networks by testing them on synthetic datasets. The datasets include common distributions of points in $R^n$ space and images containing polygons of various shapes and sizes. We find that by and large GANs fail to faithfully recreate point datasets which contain discontinous support or sharp bends with noise. Additionally, on image datasets, we find that GANs do not seem to learn to count the number of objects of the same kind in an image. We also highlight the apparent tension between generalization and learning in GANs.
Output: GANs are evaluated on synthetic datasets