Related papers: Short and Straight: Geodesics on Differentiable Ma…

We analyze a variational time discretization of geodesic calculus on finite- and certain classes of infinite-dimensional Riemannian manifolds. We investigate the fundamental properties of discrete geodesics, the associated discrete…

Random geometric graphs are random graph models defined on metric spaces. Such a model is defined by first sampling points from a metric space and then connecting each pair of sampled points with probability that depends on their distance,…

Geodesic distance matrices can reveal shape properties that are largely invariant to non-rigid deformations, and thus are often used to analyze and represent 3-D shapes. However, these matrices grow quadratically with the number of points.…

Recent literature has shown that symbolic data, such as text and graphs, is often better represented by points on a curved manifold, rather than in Euclidean space. However, geometrical operations on manifolds are generally more complicated…

Geodesic distance is the shortest path between two points in a Riemannian manifold. Manifold learning algorithms, such as Isomap, seek to learn a manifold that preserves geodesic distances. However, such methods operate on the ambient…

We propose a manifold matching approach to generative models which includes a distribution generator (or data generator) and a metric generator. In our framework, we view the real data set as some manifold embedded in a high-dimensional…

Real world data often lie on low-dimensional Riemannian manifolds embedded in high-dimensional spaces. This motivates learning degenerate normalizing flows that map between the ambient space and a low-dimensional latent space. However, if…

The geometry of generative models serves as the basis for interpolation, model inspection, and more. Unfortunately, most generative models lack a principal notion of geometry without restrictive assumptions on either the model or the data…

The minimal geodesic models based on the Eikonal equations are capable of finding suitable solutions in various image segmentation scenarios. Existing geodesic-based segmentation approaches usually exploit image features in conjunction with…

Manifold learning seeks a low dimensional representation that faithfully captures the essence of data. Current methods can successfully learn such representations, but do not provide a meaningful set of operations that are associated with…

Machine learning has been progressively generalised to operate within non-Euclidean domains, but geometrically accurate methods for learning on surfaces are still falling behind. The lack of closed-form Riemannian operators, the…

The approximation of both geodesic distances and shortest paths on point cloud sampled from an embedded submanifold $\mathcal{M}$ of Euclidean space has been a long-standing challenge in computational geometry. Given a sampling resolution…

Dataset distillation aims to synthesize a compact subset of the original data, enabling models trained on it to achieve performance comparable to those trained on the original large dataset. Existing distribution-matching methods are…

Latent space geometry provides a rigorous and empirically valuable framework for interacting with the latent variables of deep generative models. This approach reinterprets Euclidean latent spaces as Riemannian through a pull-back metric,…

Modern machine learning increasingly leverages the insight that high-dimensional data often lie near low-dimensional, non-linear manifolds, an idea known as the manifold hypothesis. By explicitly modeling the geometric structure of data…

Geodesic paths and distances are among the most popular intrinsic properties of 3D surfaces. Traditionally, geodesic paths on discrete polygon surfaces were computed using shortest path algorithms, such as Dijkstra. However, such algorithms…

A methodology is developed for data analysis based on empirically constructed geodesic metric spaces. For a probability distribution, the length along a path between two points can be defined as the amount of probability mass accumulated…

Geodesic distance, commonly called shortest path length, has proved useful in a great variety of disciplines. It has been playing a significant role in search engine at present and so attracted considerable attention at the last few…

We develop a new method for visualizing and refining the invariances of learned representations. Specifically, we test for a general form of invariance, linearization, in which the action of a transformation is confined to a low-dimensional…

For manifold learning, it is assumed that high-dimensional sample/data points are embedded on a low-dimensional manifold. Usually, distances among samples are computed to capture an underlying data structure. Here we propose a metric…