Related papers: Manifold learning in Wasserstein space
In this paper we demonstrate how sub-Riemannian geometry can be used for manifold learning and surface reconstruction by combining local linear approximations of a point cloud to obtain lower dimensional bundles. Local approximations…
We study the problem of {\em properly} learning large margin halfspaces in the agnostic PAC model. In more detail, we study the complexity of properly learning $d$-dimensional halfspaces on the unit ball within misclassification error…
We introduce a general class of transport distances ${\rm WB}_{\Lambda}$ over the space of positive semi-definite matrix-valued Radon measures $\mathcal{M}(\Omega,\mathbb{S}_+^n)$, called the weighted Wasserstein-Bures distance. Such a…
This work proposes an algorithm for explicitly constructing a pair of neural networks that linearize and reconstruct an embedded submanifold, from finite samples of this manifold. Our such-generated neural networks, called Flattening…
We investigate learning of the differential geometric structure of a data manifold embedded in a high-dimensional Euclidean space. We first analyze kernel-based algorithms and show that under the usual regularizations, non-probabilistic…
We discuss the relation between the Wasserstein distance of order 1 between probability distributions on a metric space, arising in the study of Monge-Kantorovich transport problem, and the spectral distance of noncommutative geometry.…
Manifold learning offers nonlinear dimensionality reduction of high-dimensional datasets. In this paper, we bring geometry processing to bear on manifold learning by introducing a new approach based on metric connection for generating a…
We introduce the observable Wasserstein distance, a framework for deriving lower bounds on the Wasserstein distance between probability measures on Polish metric spaces, designed to bypass the computational intractability of exact optimal…
Recently, studies on machine learning have focused on methods that use symmetry implicit in a specific manifold as an inductive bias. Grassmann manifolds provide the ability to handle fundamental shapes represented as shape spaces, enabling…
In this paper we use a diffeo-geometric framework based on manifolds that are locally modeled on "convenient" vector spaces to study the geometry of some infinite dimensional spaces. Given a finite dimensional symplectic manifold…
Understanding the space of probability measures on a metric space equipped with a Wasserstein distance is one of the fundamental questions in mathematical analysis. The Wasserstein metric has received a lot of attention in the machine…
This paper proposes two algorithms for estimating square Wasserstein distance matrices from a small number of entries. These matrices are used to compute manifold learning embeddings like multidimensional scaling (MDS) or Isomap, but…
Our world is full of asymmetries. Gravity and wind can make reaching a place easier than coming back. Social artifacts such as genealogy charts and citation graphs are inherently directed. In reinforcement learning and control, optimal…
Riemannian optimization is a principled framework for solving optimization problems where the desired optimum is constrained to a smooth manifold $\mathcal{M}$. Algorithms designed in this framework usually require some geometrical…
Gromov--Wasserstein (GW) distances compare graphs, shapes, and point clouds through internal distances, without requiring a common coordinate system. This invariance is powerful, but discrete GW is a nonconvex quadratic optimal transport…
We study the problem of PAC learning halfspaces in the reliable agnostic model of Kalai et al. (2012). The reliable PAC model captures learning scenarios where one type of error is costlier than the others. Our main positive result is a new…
In machine learning, data is usually represented in a (flat) Euclidean space where distances between points are along straight lines. Researchers have recently considered more exotic (non-Euclidean) Riemannian manifolds such as hyperbolic…
Manifold learning flows are a class of generative modelling techniques that assume a low-dimensional manifold description of the data. The embedding of such a manifold into the high-dimensional space of the data is achieved via learnable…
Manifold hypothesis states that data points in high-dimensional space actually lie in close vicinity of a manifold of much lower dimension. In many cases this hypothesis was empirically verified and used to enhance unsupervised and…
Parametrizations of data manifolds in shape spaces can be computed using the rich toolbox of Riemannian geometry. This, however, often comes with high computational costs, which raises the question if one can learn an efficient neural…