Eldad Haber

Deep neural networks have become invaluable tools for supervised machine learning, e.g., classification of text or images. While often offering superior results over traditional techniques and successfully expressing complicated patterns in data, deep architectures are known to be challenging to design and train such that they generalize well to new data. Important issues with deep architectures are numerical instabilities in derivative-based learning algorithms commonly called exploding or vanishing gradients. In this paper we propose new forward propagation techniques inspired by systems of Ordinary Differential Equations (ODE) that overcome this challenge and lead to well-posed learning problems for arbitrarily deep networks. The backbone of our approach is our interpretation of deep learning as a parameter estimation problem of nonlinear dynamical systems. Given this formulation, we analyze stability and well-posedness of deep learning and use this new understanding to develop new network architectures. We relate the exploding and vanishing gradient phenomenon to the stability of the discrete ODE and present several strategies for stabilizing deep learning for very deep networks. While our new architectures restrict the solution space, several numerical experiments show their competitiveness with state-of-the-art networks.

Deep residual networks (ResNets) and their variants are widely used in many computer vision applications and natural language processing tasks. However, the theoretical principles for designing and training ResNets are still not fully understood. Recently, several points of view have emerged to try to interpret ResNet theoretically, such as unraveled view, unrolled iterative estimation and dynamical systems view. In this paper, we adopt the dynamical systems point of view, and analyze the lesioning properties of ResNet both theoretically and experimentally. Based on these analyses, we additionally propose a novel method for accelerating ResNet training. We apply the proposed method to train ResNets and Wide ResNets for three image classification benchmarks, reducing training time by more than 40% with superior or on-par accuracy.

Deep neural networks have become invaluable tools for supervised machine learning, e.g. classification of text or images. While often offering superior results over traditional techniques and successfully expressing complicated patterns in data, deep architectures are known to be challenging to design and train such that they generalize well to new data. Critical issues with deep architectures are numerical instabilities in derivative-based learning algorithms commonly called exploding or vanishing gradients. In this paper, we propose new forward propagation techniques inspired by systems of ordinary differential equations (ODE) that overcome this challenge and lead to well-posed learning problems for arbitrarily deep networks. The backbone of our approach is our interpretation of deep learning as a parameter estimation problem of nonlinear dynamical systems. Given this formulation, we analyze stability and well-posedness of deep learning and use this new understanding to develop new network architectures. We relate the exploding and vanishing gradient phenomenon to the stability of the discrete ODE and present several strategies for stabilizing deep learning for very deep networks. While our new architectures restrict the solution space, several numerical experiments show their competitiveness with state-of-the-art networks.

Recently, deep residual networks have been successfully applied in many computer vision and natural language processing tasks, pushing the state-of-the-art performance with deeper and wider architectures. In this work, we interpret deep residual networks as ordinary differential equations (ODEs), which have long been studied in mathematics and physics with rich theoretical and empirical success. From this interpretation, we develop a theoretical framework on stability and reversibility of deep neural networks, and derive three reversible neural network architectures that can go arbitrarily deep in theory. The reversibility property allows a memory-efficient implementation, which does not need to store the activations for most hidden layers. Together with the stability of our architectures, this enables training deeper networks using only modest computational resources. We provide both theoretical analyses and empirical results. Experimental results demonstrate the efficacy of our architectures against several strong baselines on CIFAR-10, CIFAR-100 and STL-10 with superior or on-par state-of-the-art performance. Furthermore, we show our architectures yield superior results when trained using fewer training data.

We present an efficient algorithm for recent generalizations of optimal mass transport theory to matrix-valued and vector-valued densities. These generalizations lead to several applications including diffusion tensor imaging, color images processing, and multi-modality imaging. The algorithm is based on sequential quadratic programming (SQP). By approximating the Hessian of the cost and solving each iteration in an inexact manner, we are able to solve each iteration with relatively low cost while still maintaining a fast convergent rate. The core of the algorithm is solving a weighted Poisson equation, where different efficient preconditioners may be employed. We utilize incomplete Cholesky factorization, which yields an efficient and straightforward solver for our problem. Several illustrative examples are presented for both the matrix and vector-valued cases.

Full waveform inversion (FWI) is a process in which seismic numerical simulations are fit to observed data by changing the wave velocity model of the medium under investigation. The problem is non-linear, and therefore optimization techniques have been used to find a reasonable solution to the problem. The main problem in fitting the data is the lack of low spatial frequencies. This deficiency often leads to a local minimum and to non-plausible solutions. In this work we explore how to obtain low frequency information for FWI. Our approach involves augmenting FWI with travel time tomography, which has low-frequency features. By jointly inverting these two problems we enrich FWI with information that can replace low frequency data. In addition, we use high order regularization, in a preliminary inversion stage, to prevent high frequency features from polluting our model in the initial stages of the reconstruction. This regularization also promotes the non-dominant low-frequency modes that exist in the FWI sensitivity. By applying a joint FWI and travel time inversion we are able to obtain a smooth model than can later be used to recover a good approximation for the true model. A second contribution of this paper involves the acceleration of the main computational bottleneck in FWI--the solution of the Helmholtz equation. We show that the solution time can be reduced by solving the equation for multiple right hand sides using block multigrid preconditioned Krylov methods.

Estimating parameters of Partial Differential Equations (PDEs) from noisy and indirect measurements often requires solving ill-posed inverse problems. These so called parameter estimation or inverse medium problems arise in a variety of applications such as geophysical, medical imaging, and nondestructive testing. Their solution is computationally intense since the underlying PDEs need to be solved numerous times until the reconstruction of the parameters is sufficiently accurate. Typically, the computational demand grows significantly when more measurements are available, which poses severe challenges to inversion algorithms as measurement devices become more powerful. In this paper we present jInv, a flexible framework and open source software that provides parallel algorithms for solving parameter estimation problems with many measurements. Being written in the expressive programming language Julia, jInv is portable, easy to understand and extend, cross-platform tested, and well-documented. It provides novel parallelization schemes that exploit the inherent structure of many parameter estimation problems and can be used to solve multiphysics inversion problems as is demonstrated using numerical experiments motivated by geophysical imaging.

The eikonal equation is instrumental in many applications in several fields ranging from computer vision to geoscience. This equation can be efficiently solved using the iterative Fast Sweeping (FS) methods and the direct Fast Marching (FM) methods. However, when used for a point source, the original eikonal equation is known to yield inaccurate numerical solutions, because of a singularity at the source. In this case, the factored eikonal equation is often preferred, and is known to yield a more accurate numerical solution. One application that requires the solution of the eikonal equation for point sources is travel time tomography. This inverse problem may be formulated using the eikonal equation as a forward problem. While this problem has been solved using FS in the past, the more recent choice for applying it involves FM methods because of the efficiency in which sensitivities can be obtained using them. However, while several FS methods are available for solving the factored equation, the FM method is available only for the original eikonal equation. In this paper we develop a Fast Marching algorithm for the factored eikonal equation, using both first and second order finite-difference schemes. Our algorithm follows the same lines as the original FM algorithm and requires the same computational effort. In addition, we show how to obtain sensitivities using this FM method and apply travel time tomography, formulated as an inverse factored eikonal equation. Numerical results in two and three dimensions show that our algorithm solves the factored eikonal equation efficiently, and demonstrate the achieved accuracy for computing the travel time. We also demonstrate a recovery of a 2D and 3D heterogeneous medium by travel time tomography using the eikonal equation for forward modelling and inversion by Gauss-Newton.