Regularized Kernel
We introduce Regularized Kernel and Neural Sobolev Descent for transporting a source distribution to a target distribution along smooth paths of minimum kinetic energy (defined by the Sobolev discrepancy), related to dynamic optimal transport. In the kernel version, we give a simple algorithm to perform the descent along gradients of the Sobolev critic, and show that it converges asymptotically to the target distribution in the MMD sense. In the neural version, we parametrize the Sobolev critic with a neural network with input gradient norm constrained in expectation. We show in theory and experiments that regularization has an important role in favoring smooth transitions between distributions, avoiding large discrete jumps. Our analysis could provide a new perspective on the impact of critic updates (early stopping) on the paths to equilibrium in the GAN setting. …

Wasserstein Transform
We introduce the Wasserstein transform, a method for enhancing and denoising datasets defined on general metric spaces. The construction draws inspiration from Optimal Transportation ideas. We establish precise connections with the mean shift family of algorithms and establish the stability of both our method and mean shift under data perturbation. …

EnergyNet
We present ENERGYNET , a new framework for analyzing and building artificial neural network architectures. Our approach adaptively learns the structure of the networks in an unsupervised manner. The methodology is based upon the theoretical guarantees of the energy function of restricted Boltzmann machines (RBM) of infinite number of nodes. We present experimental results to show that the final network adapts to the complexity of a given problem. …

Long Term Memory Network (LTM)
Recurrent Neural Networks (RNN), Long Short-Term Memory Networks (LSTM), and Memory Networks which contain memory are popularly used to learn patterns in sequential data. Sequential data has long sequences that hold relationships. RNN can handle long sequences but suffers from the vanishing and exploding gradient problems. While LSTM and other memory networks address this problem, they are not capable of handling long sequences (50 or more data points long sequence patterns). Language modelling requiring learning from longer sequences are affected by the need for more information in memory. This paper introduces Long Term Memory network (LTM), which can tackle the exploding and vanishing gradient problems and handles long sequences without forgetting. LTM is designed to scale data in the memory and gives a higher weight to the input in the sequence. LTM avoid overfitting by scaling the cell state after achieving the optimal results. The LTM is tested on Penn treebank dataset, and Text8 dataset and LTM achieves test perplexities of 83 and 82 respectively. 650 LTM cells achieved a test perplexity of 67 for Penn treebank, and 600 cells achieved a test perplexity of 77 for Text8. LTM achieves state of the art results by only using ten hidden LTM cells for both datasets. …