Computing distances and means on manifolds with a metric-constrained Eikonal approach

• URL: http://arxiv.org/abs/2404.08754v1
• Date: Fri, 12 Apr 2024 18:26:32 GMT
• Title: Computing distances and means on manifolds with a metric-constrained Eikonal approach
• Authors: Daniel Kelshaw, Luca Magri,
• Abstract summary: We introduce the metric-constrained Eikonal solver to obtain continuous, differentiable representations of distance functions. The differentiable nature of these representations allows for the direct computation of globally length-minimising paths on the manifold.
• Score: 4.266376725904727
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Computing distances on Riemannian manifolds is a challenging problem with numerous applications, from physics, through statistics, to machine learning. In this paper, we introduce the metric-constrained Eikonal solver to obtain continuous, differentiable representations of distance functions on manifolds. The differentiable nature of these representations allows for the direct computation of globally length-minimising paths on the manifold. We showcase the use of metric-constrained Eikonal solvers for a range of manifolds and demonstrate the applications. First, we demonstrate that metric-constrained Eikonal solvers can be used to obtain the Fr\'echet mean on a manifold, employing the definition of a Gaussian mixture model, which has an analytical solution to verify the numerical results. Second, we demonstrate how the obtained distance function can be used to conduct unsupervised clustering on the manifold -- a task for which existing approaches are computationally prohibitive. This work opens opportunities for distance computations on manifolds.

Related papers

• Reconstructing the Geometry of Random Geometric Graphs [9.004991291124096]
  Random geometric graphs are random graph models defined on metric spaces. We show how to efficiently reconstruct the geometry of the underlying space from the sampled graph.
  arXiv  Detail & Related papers  (2024-02-14T21:34:44Z)
• Gaussian Entanglement Measure: Applications to Multipartite Entanglement of Graph States and Bosonic Field Theory [50.24983453990065]
  An entanglement measure based on the Fubini-Study metric has been recently introduced by Cocchiarella and co-workers. We present the Gaussian Entanglement Measure (GEM), a generalization of geometric entanglement measure for multimode Gaussian states. By providing a computable multipartite entanglement measure for systems with a large number of degrees of freedom, we show that our definition can be used to obtain insights into a free bosonic field theory.
  arXiv  Detail & Related papers  (2024-01-31T15:50:50Z)
• Scaling Riemannian Diffusion Models [68.52820280448991]
  We show that our method enables us to scale to high dimensional tasks on nontrivial manifold. We model QCD densities on $SU(n)$ lattices and contrastively learned embeddings on high dimensional hyperspheres.
  arXiv  Detail & Related papers  (2023-10-30T21:27:53Z)
• Generative Modeling on Manifolds Through Mixture of Riemannian Diffusion Processes [57.396578974401734]
  We introduce a principled framework for building a generative diffusion process on general manifold. Instead of following the denoising approach of previous diffusion models, we construct a diffusion process using a mixture of bridge processes. We develop a geometric understanding of the mixture process, deriving the drift as a weighted mean of tangent directions to the data points.
  arXiv  Detail & Related papers  (2023-10-11T06:04:40Z)
• Manifold-augmented Eikonal Equations: Geodesic Distances and Flows on Differentiable Manifolds [5.0401589279256065]
  We show how the geometry of a manifold impacts the distance field, and exploit the geodesic flow to obtain globally length-minimising curves directly. This work opens opportunities for statistics and reduced-order modelling on differentiable manifold.
  arXiv  Detail & Related papers  (2023-10-09T21:11:13Z)
• Short and Straight: Geodesics on Differentiable Manifolds [6.85316573653194]
  In this work, we first analyse existing methods for computing length-minimising geodesics. Second, we propose a model-based parameterisation for distance fields and geodesic flows on continuous manifold. Third, we develop a curvature-based training mechanism, sampling and scaling points in regions of the manifold exhibiting larger values of the Ricci scalar.
  arXiv  Detail & Related papers  (2023-05-24T15:09:41Z)
• Manifold Learning by Mixture Models of VAEs for Inverse Problems [1.5749416770494704]
  We learn a mixture model of variational autoencoders to represent a manifold of arbitrary topology. We use it for solving inverse problems by minimizing a data fidelity term restricted to the learned manifold. We demonstrate the performance of our method for low-dimensional toy examples as well as for deblurring and electrical impedance tomography.
  arXiv  Detail & Related papers  (2023-03-27T14:29:04Z)
• Convolutional Filtering on Sampled Manifolds [122.06927400759021]
  We show that convolutional filtering on a sampled manifold converges to continuous manifold filtering. Our findings are further demonstrated empirically on a problem of navigation control.
  arXiv  Detail & Related papers  (2022-11-20T19:09:50Z)
• Bayesian Quadrature on Riemannian Data Manifolds [79.71142807798284]
  A principled way to model nonlinear geometric structure inherent in data is provided. However, these operations are typically computationally demanding. In particular, we focus on Bayesian quadrature (BQ) to numerically compute integrals over normal laws. We show that by leveraging both prior knowledge and an active exploration scheme, BQ significantly reduces the number of required evaluations.
  arXiv  Detail & Related papers  (2021-02-12T17:38:04Z)
• Intrinsic persistent homology via density-based metric learning [1.0499611180329804]
  We prove that the metric space defined by the sample endowed with a computable metric known as sample Fermat distance converges a.s. The limiting object is the manifold itself endowed with the population Fermat distance, an intrinsic metric that accounts for both the geometry of the manifold and the density that produces the sample.
  arXiv  Detail & Related papers  (2020-12-11T18:54:36Z)
• Disentangling by Subspace Diffusion [72.1895236605335]
  We show that fully unsupervised factorization of a data manifold is possible if the true metric of the manifold is known. Our work reduces the question of whether unsupervised metric learning is possible, providing a unifying insight into the geometric nature of representation learning.
  arXiv  Detail & Related papers  (2020-06-23T13:33:19Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.