Related papers: Simultaneous Optimization of Geodesics and Fr\'ech…
Given two simplicial complexes in R^d, and start and end vertices in each complex, we show how to compute curves (in each complex) between these vertices, such that the Fr\'echet distance between these curves is minimized. As a polygonal…
Local Fr\'echet regression is a nonparametric regression method for metric space valued responses and Euclidean predictors, which can be utilized to obtain estimates of smooth trajectories taking values in general metric spaces from noisy…
The geometric median as well as the Frechet mean of points in an Hadamard space are important in both theory and applications. Surprisingly, no algorithms for their computation are hitherto known. To address this issue, we use a split…
Stochastic variance reduction algorithms have recently become popular for minimizing the average of a large, but finite, number of loss functions. In this paper, we propose a novel Riemannian extension of the Euclidean stochastic variance…
Riemannian geometry provides us with powerful tools to explore the latent space of generative models while preserving the underlying structure of the data. The latent space can be equipped it with a Riemannian metric, pulled back from the…
We develop a geometric convergence theory for neural-network optimization within the minimizing movement scheme (MMS) framework. Reformulating each neural MMS step as a minimization over the set of increments in a Hilbert space, we show…
For most optimisation methods an essential assumption is the vector space structure of the feasible set. This condition is not fulfilled if we consider optimisation problems over the sphere. We present an algorithm for solving a special…
We study the problem of network regression, where one is interested in how the topology of a network changes as a function of Euclidean covariates. We build upon recent developments in generalized regression models on metric spaces based on…
The techniques and analysis presented in this thesis provide new methods to solve optimization problems posed on Riemannian manifolds. These methods are applied to the subspace tracking problem found in adaptive signal processing and…
This paper considers the problem of finding a meaningful template function that represents the common pattern of a sample of curves. To address this issue, a novel algorithm based on a robust version of the isometric featuring mapping…
Inferring time-varying graph structures from high-dimensional nodal observations is a fundamental problem arising in neuroscience, finance, climatology, and beyond. Two intrinsic challenges govern this problem: maintaining the…
Manifolds discovered by machine learning models provide a compact representation of the underlying data. Geodesics on these manifolds define locally length-minimising curves and provide a notion of distance, which are key for reduced-order…
We provide two closed-form geodesic formulas for a family of metrics on Stiefel manifold, parameterized by two positive numbers, having both the embedded and canonical metrics as special cases. The closed-form formulas allow us to compute…
Many geometric machine learning and image analysis applications, require a left-invariant metric on the 5D homogeneous space of 3D positions and orientations SE(3)/SO(2). This is done in Equivariant Neural Networks (G-CNNs), or in PDE-Based…
In this paper, we develop a novel method for fast geodesic distance queries. The key idea is to embed the mesh into a high-dimensional space, such that the Euclidean distance in the high-dimensional space can induce the geodesic distance in…
This paper is devoted to the statistical and numerical properties of the geometric median, and its applications to the problem of robust mean estimation via the median of means principle. Our main theoretical results include (a) an upper…
Geometric frameworks for analyzing curves are common in applications as they focus on invariant features and provide visually satisfying solutions to standard problems such as computing invariant distances, averaging curves, or registering…
Fr\'echet means are a popular type of average for non-Euclidean datasets, defined as those points which minimise the average squared distance to a set of data points. We consider the behaviour of sample Fr\'echet means on normed spaces…
Mixed effect modeling for longitudinal data is challenging when the observed data are random objects, which are complex data taking values in a general metric space without linear structure. In such settings the classical additive error…
Over the past few years, symmetric positive definite (SPD) matrices have been receiving considerable attention from computer vision community. Though various distance measures have been proposed in the past for comparing SPD matrices, the…