Related papers: Geodesics in the Deep Linear Network

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…

This paper investigates sub-Riemannian geodesics within the jet space of curves. We establish the existence of two distinct families of metric lines, that is, globally minimizing geodesics, in the $2$-jet space of plane curves. This result…

There are many surprising and perhaps counter-intuitive properties of optimization of deep neural networks. We propose and experimentally verify a unified phenomenological model of the loss landscape that incorporates many of them. High…

We explore a nonlocal connection between certain linear and nonlinear ordinary differential equations (ODEs), representing physically important oscillator systems, and identify a class of integrable nonlinear ODEs of any order. We also…

We introduce a class of special geometries associated to the choice of a differential graded algebra contained in \Lambda R^n. We generalize some known embedding results, that effectively characterize the real analytic Riemannian manifolds…

Modern machine learning increasingly leverages the insight that high-dimensional data often lie near low-dimensional, non-linear manifolds, an idea known as the manifold hypothesis. By explicitly modeling the geometric structure of data…

Suppose that there exists a discrete subset $X$ of a complete, connected, $n$-dimensional Riemannian manifold $M$ such that the Riemannian distances between points of $X$ correspond to the Euclidean distances of a net in $\mathbb{R}^{n}$.…

The problem of recovering the configuration of points from their partial pairwise distances, referred to as the Euclidean Distance Matrix Completion (EDMC) problem, arises in a broad range of applications, including sensor network…

We give a natural definition of geodesics on a Riemannian supermanifold and extend the usual geodesic flow defined on the cotangent bundle of the body of the supermanifold, associated to the induced Riemannian structure on the body, to a…

We suggest a universal map capable to recover a behavior of a wide range of dynamical systems given by ODEs. The map is built as an artificial neural network whose weights encode a modeled system. We assume that ODEs are known and prepare…

The two-dimensional surface of a bi-axial ellipsoid is characterized by the lengths of its major and minor axes. Longitude and latitude span an angular coordinate system across. We consider the egg-shaped surface of constant altitude above…

We introduce the geodesic walk for sampling Riemannian manifolds and apply it to the problem of generating uniform random points from polytopes in R^n specified by m inequalities. The walk is a discrete-time simulation of a stochastic…

Deep neural networks (DNNs) on Riemannian manifolds have garnered increasing interest in various applied areas. For instance, DNNs on spherical and hyperbolic manifolds have been designed to solve a wide range of computer vision and nature…

Recent work in deep learning focuses on solving physical systems in the Ordinary Differential Equation or Partial Differential Equation. This current work proposed a variant of Convolutional Neural Networks (CNNs) that can learn the hidden…

Machine learning methods are commonly used to solve inverse problems, wherein an unknown signal must be estimated from few indirect measurements generated via a known acquisition procedure. In particular, neural networks perform well…

We consider the leapfrog algorithm by Noakes for computing geodesics on Riemannian manifolds. The main idea behind this algorithm is to subdivide the original endpoint geodesic problem into several local problems, for which the endpoint…

We begin by characterizing metabelian distributions in terms of principal bundle structures. Then, we prove that in sub-Riemannian manifolds with metabelian distributions of rank $r$, the projection of strictly singular trajectories to some…

Residual networks (ResNets) have displayed impressive results in pattern recognition and, recently, have garnered considerable theoretical interest due to a perceived link with neural ordinary differential equations (neural ODEs). This link…

The paper studies the harmonic maps on a direction between a Riemannian space and a generalized Lagrange space. Also, it is proved there that the solutions of C^2 class of certain ODEs or PDEs are harmonic maps, in the sense of this paper.

We introduce anti-invariant Riemannian submersions from Sasakian manifolds onto Riemannian manifolds. We survey main results of anti-invariant Riemannian submersions defined on Sasakian manifolds. We investigate necessary and sufficient…