Related papers: Riemannian Convex Potential Maps
Rotation vectors, as defined for homeomorphisms of the torus that are isotopic to the identity, are generalized to such homeomorphisms of any complete Riemannian manifold with non-positive sectional curvature. These generalized rotation…
In this thesis, we study extensions of the theory of Riemannian submanifolds in two directions. First, we will show how Riemannian geometry and submanifold theory in particular, can be generalized using the notion of 'Rinehart spaces', and…
We consider optimization problems on manifolds with equality and inequality constraints. A large body of work treats constrained optimization in Euclidean spaces. In this work, we consider extensions of existing algorithms from the…
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…
Normalizing flows are a promising tool for modeling probability distributions in physical systems. While state-of-the-art flows accurately approximate distributions and energies, applications in physics additionally require smooth energies…
In this paper, we derive a number of interesting properties and extensions of the convex flow problem from the perspective of convex geometry. We show that the sets of allowable flows always can be imbued with a downward closure property,…
In this paper, we address the problem of optimizing flows on generalized graphs that feature multiple entry points and multiple populations, each with varying cost structures. We tackle this problem by considering the multi-population…
Circular and non-flat data distributions are prevalent across diverse domains of data science, yet their specific geometric structures often remain underutilized in machine learning frameworks. A principled approach to accounting for the…
This paper proposes an original Riemmanian geometry for low-rank structured elliptical models, i.e., when samples are elliptically distributed with a covariance matrix that has a low-rank plus identity structure. The considered geometry is…
Many machine learning problems involve data supported on curved spaces such as spheres, rotation groups, hyperbolic spaces, and general Riemannian manifolds, where Euclidean geometry can distort distances, averages, and the resulting…
The elastic flow, which is the $L^2$-gradient flow of the elastic energy, has several applications in geometry and elasticity theory. We present stable discretizations for the elastic flow in two-dimensional Riemannian manifolds that are…
In this paper we bring together some of the key ideas and methods of two disparate fields of mathematical research, frame theory and optimal transport, using the methods of the second to answer questions posed in the first. In particular,…
Persistence diagrams (PDs) are now routinely used to summarize the underlying topology of complex data. Despite several appealing properties, incorporating PDs in learning pipelines can be challenging because their natural geometry is not…
We further research on the accelerated optimization phenomenon on Riemannian manifolds by introducing accelerated global first-order methods for the optimization of $L$-smooth and geodesically convex (g-convex) or $\mu$-strongly g-convex…
Recent advances have allowed to tackle exact path-space probabilistic representations of macroscopic advection-diffusion models involving advection nonlinearities by step forward approaches in terms of continuous branching stochastic…
In this paper, we establish, for the first time, upper bounds of the Ricci--curvature for Riemannian submersions along the vertical distribution as well as along both the vertical and horizontal distributions. We derive their general forms…
This paper studies large-scale optimization problems on Riemannian manifolds whose objective function is a finite sum of negative log-probability losses. Such problems arise in various machine learning and signal processing applications. By…
Optimization with orthogonality constraints frequently arises in various fields such as machine learning. Riemannian optimization offers a powerful framework for solving these problems by equipping the constraint set with a Riemannian…
We introduce the convex bundle method to solve convex, non-smooth optimization problems on Riemannian manifolds of bounded sectional curvature. Each step of our method is based on a model that involves the convex hull of previously…
In this manuscript we review new ideas and first results on application of the Graphical Models approach, originated from Statistical Physics, Information Theory, Computer Science and Machine Learning, to optimization problems of network…