相关论文: Denoising distances beyond the volumetric barrier

The maximum mean discrepancy and Wasserstein distance are popular distance measures between distributions and play important roles in many machine learning problems such as metric learning, generative modeling, domain adaption, and…

Estimating Wasserstein distances between two high-dimensional densities suffers from the curse of dimensionality: one needs an exponential (wrt dimension) number of samples to ensure that the distance between two empirical measures is…

We propose an inexact optimization algorithm on Riemannian manifolds, motivated by quadratic discrimination tasks in high-dimensional, low-sample-size (HDLSS) imaging settings. In such applications, gradient evaluations are often biased due…

The Gromov--Wasserstein (GW) distance and its fused extension (FGW) are powerful tools for comparing heterogeneous data. Their computation is, however, challenging since both distances are based on non-convex, quadratic optimal transport…

Existing approaches to depth or disparity estimation output a distribution over a set of pre-defined discrete values. This leads to inaccurate results when the true depth or disparity does not match any of these values. The fact that this…

Gromov-Wasserstein distance has found many applications in machine learning due to its ability to compare measures across metric spaces and its invariance to isometric transformations. However, in certain applications, this invariance…

In this article, we derive Stein's method for approximating a spatial random graph by a generalised random geometric graph, which has vertices given by a finite Gibbs point process and edges based on a general connection function. Our main…

This paper considers the problem of regression over distributions, which is becoming increasingly important in machine learning. Existing approaches often ignore the geometry of the probability space or are computationally expensive. To…

A common observation in data-driven applications is that high-dimensional data have a low intrinsic dimension, at least locally. In this work, we consider the problem of point estimation for manifold-valued data. Namely, given a finite set…

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…

We consider the problem of reconstructing an embedding of a compact connected Riemannian manifold in a Euclidean space up to an almost isometry, given the information on intrinsic distances between points from its ``sufficiently large''…

Given dense image feature correspondences of a non-rigidly moving object across multiple frames, this paper proposes an algorithm to estimate its 3D shape for each frame. To solve this problem accurately, the recent state-of-the-art…

We study the recovery of geometric structure from data generated by convolving the uniform measure on a smooth compact submanifold $M\subset\mathbb{R}^D$ with ambient Gaussian noise. Our main result is that several fundamental Riemannian…

We demonstrate a simple greedy algorithm that can reliably recover a d-dimensional vector v from incomplete and inaccurate measurements x. Here our measurement matrix is an N by d matrix with N much smaller than d. Our algorithm,…

This paper concerns the problem of recovering an unknown but structured signal $x \in R^n$ from $m$ quadratic measurements of the form $y_r=|<a_r,x>|^2$ for $r=1,2,...,m$. We focus on the under-determined setting where the number of…

We investigate the geometrical structure of probabilistic generative dimensionality reduction models using the tools of Riemannian geometry. We explicitly define a distribution over the natural metric given by the models. We provide the…

In this work, we present a method to compute the Kantorovich-Wasserstein distance of order one between a pair of two-dimensional histograms. Recent works in Computer Vision and Machine Learning have shown the benefits of measuring…

This paper studies the problem of recovering the hidden vertex correspondence between two edge-correlated random graphs. We focus on the Gaussian model where the two graphs are complete graphs with correlated Gaussian weights and the…

It has long been thought that high-dimensional data encountered in many practical machine learning tasks have low-dimensional structure, i.e., the manifold hypothesis holds. A natural question, thus, is to estimate the intrinsic dimension…

In this paper, we leverage the properties of non-Euclidean Geometry to define the Geodesic distance (GD) on the space of statistical manifolds. The Geodesic distance is a real and intuitive similarity measure that is a good alternative to…