Turing approximations, toric isometric embeddings & manifold convolutions

• URL: http://arxiv.org/abs/2110.02279v1
• Date: Tue, 5 Oct 2021 18:36:16 GMT
• Title: Turing approximations, toric isometric embeddings & manifold convolutions
• Authors: P. Su\'arez-Serrato
• Abstract summary: We define a convolution operator for a manifold of arbitrary topology and dimension. A result of Alan Turing from 1938 underscores the need for such a toric isometric embedding approach to achieve a global definition of convolution.
• Score: 0.0
• License: http://creativecommons.org/licenses/by-nc-nd/4.0/
• Abstract: Convolutions are fundamental elements in deep learning architectures. Here, we present a theoretical framework for combining extrinsic and intrinsic approaches to manifold convolution through isometric embeddings into tori. In this way, we define a convolution operator for a manifold of arbitrary topology and dimension. We also explain geometric and topological conditions that make some local definitions of convolutions which rely on translating filters along geodesic paths on a manifold, computationally intractable. A result of Alan Turing from 1938 underscores the need for such a toric isometric embedding approach to achieve a global definition of convolution on computable, finite metric space approximations to a smooth manifold.

Related papers

• 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)
• Toward a Geometric Theory of Manifold Untangling [12.229735866373073]
  We conjecture that there is a more general solution to manifold untangling in the topological space without artificially defining any distance metric. General strategies of both global manifold embedding and local manifold flattening are presented and connected with existing work on the untangling of image, audio, and language data.
  arXiv  Detail & Related papers  (2023-03-07T19:47:01Z)
• 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)
• Dist2Cycle: A Simplicial Neural Network for Homology Localization [66.15805004725809]
  Simplicial complexes can be viewed as high dimensional generalizations of graphs that explicitly encode multi-way ordered relations. We propose a graph convolutional model for learning functions parametrized by the $k$-homological features of simplicial complexes.
  arXiv  Detail & Related papers  (2021-10-28T14:59:41Z)
• The decomposition of the higher-order homology embedding constructed from the $k$-Laplacian [5.076419064097734]
  The null space of the $k$-th order Laplacian $mathbfmathcal L_k$ encodes the non-trivial topology of a manifold or a network. We propose an algorithm to factorize the homology embedding into subspaces corresponding to a manifold's simplest topological components.
  arXiv  Detail & Related papers  (2021-07-23T00:40:01Z)
• Improving Metric Dimensionality Reduction with Distributed Topology [68.8204255655161]
  DIPOLE is a dimensionality-reduction post-processing step that corrects an initial embedding by minimizing a loss functional with both a local, metric term and a global, topological term. We observe that DIPOLE outperforms popular methods like UMAP, t-SNE, and Isomap on a number of popular datasets.
  arXiv  Detail & Related papers  (2021-06-14T17:19:44Z)
• A Unifying and Canonical Description of Measure-Preserving Diffusions [60.59592461429012]
  A complete recipe of measure-preserving diffusions in Euclidean space was recently derived unifying several MCMC algorithms into a single framework. We develop a geometric theory that improves and generalises this construction to any manifold.
  arXiv  Detail & Related papers  (2021-05-06T17:36:55Z)
• Manifold learning with arbitrary norms [8.433233101044197]
  We show that manifold learning based on Earthmover's distances outperforms the standard Euclidean variant for learning molecular shape spaces. We show in a numerical simulation that manifold learning based on Earthmover's distances outperforms the standard Euclidean variant for learning molecular shape spaces.
  arXiv  Detail & Related papers  (2020-12-28T10:24:30Z)
• Orthologics for Cones [5.994412766684843]
  In this paper we study logics for such geometric structures. We describe an extension of minimal orthologic with a partial modularity rule that holds for closed convex cones. This logic combines a feasible data structure (exploiting convexity/conicity) with sufficient expressivity, including full orthonegation.
  arXiv  Detail & Related papers  (2020-08-07T13:28:27Z)
• 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)
• Neural Ordinary Differential Equations on Manifolds [0.342658286826597]
  Recently normalizing flows in Euclidean space based on Neural ODEs show great promise, yet suffer the same limitations. We show how vector fields provide a general framework for parameterizing a flexible class of invertible mapping on these spaces.
  arXiv  Detail & Related papers  (2020-06-11T17:56:34Z)

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.