Related papers: Manifold-augmented Eikonal Equations: Geodesic Dis…

This paper presents mathematical results in support of the methodology of the probabilistic learning on manifolds (PLoM) recently introduced by the authors, which has been used with success for analyzing complex engineering systems. The…

This paper presents a mathematical framework for analyzing machine learning models through the geometry of their induced partitions. By representing partitions as Riemannian simplicial complexes, we capture not only adjacency relationships…

Probabilistic Latent Variable Models (LVMs) excel at modeling complex, high-dimensional data through lower-dimensional representations. Recent advances show that equipping these latent representations with a Riemannian metric unlocks…

Quantum states defined over a parameter space form a Grassmann manifold. To capture the geometry of the associated gauge structure, gauge-invariant quantities are essential. We employ the projector of a multilevel system to quantify the…

We compute the length of geodesics on a Riemannian manifold by regular polynomial interpolation of the global solution of the eikonal equation related to the line element $ds^2=g_{ij}dx^idx^j$ of the manifold. Our algorithm approximates the…

Neural models learn representations of high-dimensional data on low-dimensional manifolds. Multiple factors, including stochasticities in the training process, model architectures, and additional inductive biases, may induce different…

Riemannian optimization uses local methods to solve optimization problems whose constraint set is a smooth manifold. A linear step along some descent direction usually leaves the constraints, and hence retraction maps are used to…

Despite high-dimensionality of images, the sets of images of 3D objects have long been hypothesized to form low-dimensional manifolds. What is the nature of such manifolds? How do they differ across objects and object classes? Answering…

Manifold-learning techniques are routinely used in mining complex spatiotemporal data to extract useful, parsimonious data representations/parametrizations; these are, in turn, useful in nonlinear model identification tasks. We focus here…

Metrics on Lie groupoids and differentiable stacks have been introduced recently, extending the Riemannian geometry of manifolds and orbifolds to more general singular spaces. Here we continue that theory, studying stacky curves on…

In this paper, we study linear regression applied to data structured on a manifold. We assume that the data manifold is smooth and is embedded in a Euclidean space, and our objective is to reveal the impact of the data manifold's extrinsic…

Geodesics on Riemannian manifolds are precisely the locally length-minimizing curves, but their explicit description via simple functions is rarely possible. Geodesics of the simplest form, such as lines on Euclidean space and great circles…

We determine the Euclidean distance degrees of the three most common manifolds arising in manifold optimization: flag, Grassmann, and Stiefel manifolds. For the Grassmannian, we will also determine the Euclidean distance degree of an…

In this paper we obtain estimates on the distance of inertial manifolds for dynamical systems generated by evolutionary parabolic type equations. We consider the situation where the systems are defined in different phase spaces and we…

We consider the evolution of a compact segment of an analytic curve on the unit tangent bundle of a finite volume hyperbolic $n$-manifold under the geodesic flow. Suppose that the curve is not contained in a stable leaf of the flow. It is…

Finding meaningful representations and distances of hierarchical data is important in many fields. This paper presents a new method for hierarchical data embedding and distance. Our method relies on combining diffusion geometry, a central…

In this paper, we propose the use of geodesic distances in conjunction with multivariate distance matrix regression, called geometric-MDMR, as a powerful first step analysis method for manifold-valued data. Manifold-valued data is appearing…

Although Deep Learning (DL) has achieved success in complex Artificial Intelligence (AI) tasks, it suffers from various notorious problems (e.g., feature redundancy, and vanishing or exploding gradients), since updating parameters in…

Although many machine learning algorithms involve learning subspaces with particular characteristics, optimizing a parameter matrix that is constrained to represent a subspace can be challenging. One solution is to use Riemannian…

This paper investigates the relationship between a system of differential equations and the underlying geometry associated with it. The geometry of a surface determines shortest paths, or geodesics connecting nearby points, which are…