Time-lagged autoencoders: Deep learning of slow collective variables for molecular kinetics

Inspired by the success of deep learning techniques in the physical and chemical sciences, we apply a modification of an autoencoder type deep neural network to the task of dimension reduction of molecular dynamics data. We can show that our time-lagged autoencoder reliably finds low-dimensional embeddings for high-dimensional feature spaces which capture the slow dynamics of the underlying stochastic processes - beyond the capabilities of linear dimension reduction techniques

Finding Critical and Gradient-Flat Points of Deep Neural Network Loss Functions

Despite the fact that the loss functions of deep neural networks are highly non-convex, gradient-based optimization algorithms converge to approximately the same performance from many random initial points. This makes neural networks easy to train, which, combined with their high representational capacity and implicit and explicit regularization strategies, leads to machine-learned algorithms of high quality with reasonable computational cost in a wide variety of domains. One thread of work has focused on explaining this phenomenon by numerically characterizing the local curvature at critical points of the loss function, where gradients are zero. Such studies have reported that the loss functions used to train neural networks have no local minima that are much worse than global minima, backed up by arguments from random matrix theory. More recent theoretical work, however, has suggested that bad local minima do exist. In this dissertation, we show that one cause of this gap is that the methods used to numerically find critical points of neural network losses suffer, ironically, from a bad local minimum problem of their own. This problem is caused by gradient-flat points, where the gradient vector is in the kernel of the Hessian matrix of second partial derivatives. At these points, the loss function becomes, to second order, linear in the direction of the gradient, which violates the assumptions necessary to guarantee convergence for second order critical point-finding methods. We present evidence that approximately gradient-flat points are a common feature of several prototypical neural network loss functions

MHR-Net: Multiple-Hypothesis Reconstruction of Non-Rigid Shapes from 2D Views

We propose MHR-Net, a novel method for recovering Non-Rigid Shapes from Motion (NRSfM). MHR-Net aims to find a set of reasonable reconstructions for a 2D view, and it also selects the most likely reconstruction from the set. To deal with the challenging unsupervised generation of non-rigid shapes, we develop a new Deterministic Basis and Stochastic Deformation scheme in MHR-Net. The non-rigid shape is first expressed as the sum of a coarse shape basis and a flexible shape deformation, then multiple hypotheses are generated with uncertainty modeling of the deformation part. MHR-Net is optimized with reprojection loss on the basis and the best hypothesis. Furthermore, we design a new Procrustean Residual Loss, which reduces the rigid rotations between similar shapes and further improves the performance. Experiments show that MHR-Net achieves state-of-the-art reconstruction accuracy on Human3.6M, SURREAL and 300-VW datasets.Comment: Accepted to ECCV 202