Search

We propose a simple but effective modification of the discriminators, namely measure-conditional discriminators, as a plug-and-play module for different GANs. By taking the generated distributions as part of input so that the target optimum for the discriminator is stationary, the proposed discriminator is more robust than the vanilla one. A variant of the measure-conditional discriminator can also handle multiple target distributions, or act as a surrogate model of statistical distances such as KL divergence with applications to transfer learning.

We propose novel connections between several neural network architectures and viscosity solutions of some Hamilton--Jacobi (HJ) partial differential equations (PDEs) whose Hamiltonian is convex and only depends on the spatial gradient of the solution. To be specific, we prove that under certain assumptions, the two neural network architectures we proposed represent viscosity solutions to two sets of HJ PDEs with zero error. We also implement our proposed neural network architectures using Tensorflow and provide several examples and illustrations. Note that these neural network representations can avoid curve of dimensionality for certain HJ PDEs, since they do not involve neither grids nor discretization. Our results suggest that efficient dedicated hardware implementation for neural networks can be leveraged to evaluate viscosity solutions of certain HJ PDEs.

This paper provides new theoretical connections between multi-time Hamilton-Jacobi partial differential equations and variational image decomposition models in imaging sciences. We show that the minimal values of these optimization problems are governed by multi-time Hamilton-Jacobi partial differential equations. The minimizers of these optimization problems can be represented using the momentum in the corresponding Hamilton-Jacobi partial differential equation. Moreover, variational behaviors of both the minimizers and the momentum are investigated as the regularization parameters approach zero. In addition, we provide a new perspective from convex analysis to prove the uniqueness of convex solutions to Hamilton-Jacobi equations. Finally we consider image decomposition models that do not have unique minimizers and we propose a regularization approach to perform the analysis using multi-time Hamilton-Jacobi partial differential equations.