Short and Straight: Geodesics on Differentiable Manifolds

• URL: http://arxiv.org/abs/2305.15228v1
• Date: Wed, 24 May 2023 15:09:41 GMT
• Title: Short and Straight: Geodesics on Differentiable Manifolds
• Authors: Daniel Kelshaw, Luca Magri
• Abstract summary: 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.
• Score: 6.85316573653194
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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 modelling, statistical inference, and interpolation. In this work, we first analyse existing methods for computing length-minimising geodesics. We find that these are not suitable for obtaining valid paths, and thus, geodesic distances. We remedy these shortcomings by leveraging numerical tools from differential geometry, which provide the means to obtain Hamiltonian-conserving geodesics. Second, we propose a model-based parameterisation for distance fields and geodesic flows on continuous manifolds. Our approach exploits a manifold-aware extension to the Eikonal equation, eliminating the need for approximations or discretisation. Finally, we develop a curvature-based training mechanism, sampling and scaling points in regions of the manifold exhibiting larger values of the Ricci scalar. This sampling and scaling approach ensures that we capture regions of the manifold subject to higher degrees of geodesic deviation. Our proposed methods provide principled means to compute valid geodesics and geodesic distances on manifolds. This work opens opportunities for latent-space interpolation, optimal control, and distance computation on differentiable manifolds.

Related papers

• 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)
• 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)
• Warped geometric information on the optimisation of Euclidean functions [43.43598316339732]
  We consider optimisation of a real-valued function defined in a potentially high-dimensional Euclidean space. We find the function's optimum along a manifold with a warped metric. Our proposed algorithm, using 3rd-order approximation of geodesics, tends to outperform standard Euclidean gradient-based counterparts.
  arXiv  Detail & Related papers  (2023-08-16T12:08:50Z)
• Curvature-Independent Last-Iterate Convergence for Games on Riemannian Manifolds [77.4346324549323]
  We show that a step size agnostic to the curvature of the manifold achieves a curvature-independent and linear last-iterate convergence rate. To the best of our knowledge, the possibility of curvature-independent rates and/or last-iterate convergence has not been considered before.
  arXiv  Detail & Related papers  (2023-06-29T01:20:44Z)
• A Heat Diffusion Perspective on Geodesic Preserving Dimensionality Reduction [66.21060114843202]
  We propose a more general heat kernel based manifold embedding method that we call heat geodesic embeddings. Results show that our method outperforms existing state of the art in preserving ground truth manifold distances. We also showcase our method on single cell RNA-sequencing datasets with both continuum and cluster structure.
  arXiv  Detail & Related papers  (2023-05-30T13:58:50Z)
• VTAE: Variational Transformer Autoencoder with Manifolds Learning [144.0546653941249]
  Deep generative models have demonstrated successful applications in learning non-linear data distributions through a number of latent variables. The nonlinearity of the generator implies that the latent space shows an unsatisfactory projection of the data space, which results in poor representation learning. We show that geodesics and accurate computation can substantially improve the performance of deep generative models.
  arXiv  Detail & Related papers  (2023-04-03T13:13:19Z)
• Information Entropy Initialized Concrete Autoencoder for Optimal Sensor Placement and Reconstruction of Geophysical Fields [58.720142291102135]
  We propose a new approach to the optimal placement of sensors for reconstructing geophysical fields from sparse measurements. We demonstrate our method on the two examples: (a) temperature and (b) salinity fields around the Barents Sea and the Svalbard group of islands. We find out that the obtained optimal sensor locations have clear physical interpretation and correspond to the boundaries between sea currents.
  arXiv  Detail & Related papers  (2022-06-28T12:43:38Z)
• GraphWalks: Efficient Shape Agnostic Geodesic Shortest Path Estimation [93.60478281489243]
  We propose a learnable network to approximate geodesic paths on 3D surfaces. The proposed method provides efficient approximations of the shortest paths and geodesic distances estimations.
  arXiv  Detail & Related papers  (2022-05-30T16:22:53Z)
• Geodesic Models with Convexity Shape Prior [8.932981695464761]
  In this paper, we take into account a more complicated problem: finding curvature-penalized geodesic paths with a convexity shape prior. We establish new geodesic models relying on the strategy of orientation-lifting. The convexity shape prior serves as a constraint for the construction of local geodesic metrics encoding a curvature constraint.
  arXiv  Detail & Related papers  (2021-11-01T09:41:54Z)
• A Graph-based approach to derive the geodesic distance on Statistical manifolds: Application to Multimedia Information Retrieval [5.1388648724853825]
  We leverage the properties of non-Euclidean Geometry to define the Geodesic distance. We propose an approximation of the Geodesic distance through a graph-based method. Our main aim is to compare the graph-based approximation to the state of the art approximations.
  arXiv  Detail & Related papers  (2021-06-26T16:39:54Z)
• Uniform Interpolation Constrained Geodesic Learning on Data Manifold [28.509561636926414]
  Along the learned geodesic, our method can generate high-qualitys between two given data samples. We provide a theoretical analysis of our model and use image translation as an example to demonstrate the effectiveness of our method.
  arXiv  Detail & Related papers  (2020-02-12T07:47:41Z)

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.