mlsanityData to 7 Oct 2026

Fully-Connected Network on Noncompact Symmetric Space and Ridgelet Transform based on Helgason-Fourier Analysis

arXiv 2203.01631 ↗First posted 3 Mar 2022cs.LGcited by 14

Not on the map. Why?

Abstract (arXiv)

Neural network on Riemannian symmetric space such as hyperbolic space and the manifold of symmetric positive definite (SPD) matrices is an emerging subject of research in geometric deep learning. Based on the well-established framework of the Helgason-Fourier transform on the noncompact symmetric space, we present a fully-connected network and its associated ridgelet transform on the noncompact symmetric space, covering the hyperbolic neural network (HNN) and the SPDNet as special cases. The ridgelet transform is an analysis operator of a depth-2 continuous network spanned by neurons, namely, it maps an arbitrary given function to the weights of a network. Thanks to the coordinate-free reformulation, the role of nonlinear activation functions is revealed to be a wavelet function, and the reconstruction formula directly yields the universality of the proposed networks.

Key facts. Cited by 14 papers in the map (data through 7 Oct 2026). Builds on 24 of its 67 references that are in the map. First posted on arXiv 3 Mar 2022.

Builds on

24 of 67 references.

References · 24

  1. 2021Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gaugescited by 1,131
  2. 2017Poincar'e Embeddings for Learning Hierarchical Representationscited by 674
  3. 2018A Mean Field View of the Landscape of Two-Layers Neural Networkscited by 463
  4. 2018On the Global Convergence of Gradient Descent for Over-parameterized Models using Optimal Transportcited by 422
  5. 2018Hyperbolic Neural Networkscited by 393
  6. 2018Learning Continuous Hierarchies in the Lorentz Model of Hyperbolic Geometrycited by 261
  7. 2018Representation Tradeoffs for Hyperbolic Embeddingscited by 204
  8. 2016A Riemannian Network for SPD Matrix Learningcited by 162
  9. 2018Hyperbolic Attention Networkscited by 122
  10. 2019A Function Space View of Bounded Norm Infinite Width ReLU Nets: The Multivariate Casecited by 113
  11. 2019How do infinite width bounded norm networks look in function space?cited by 105
  12. 2019Riemannian batch normalization for SPD neural networkscited by 72
12 more
  1. 2015Neural Network with Unbounded Activation Functions is Universal Approximatorcited by 63
  2. 2017Stochastic Particle Gradient Descent for Infinite Ensemblescited by 52
  3. 2018A representer theorem for deep neural networkscited by 38
  4. 2018Equivalence of approximation by convolutional neural networks and fully-connected networkscited by 38
  5. 2020Computationally Tractable Riemannian Manifolds for Graph Embeddingscited by 29
  6. 2021Vector-valued Distance and Gyrocalculus on the Space of Symmetric Positive Definite Matricescited by 22
  7. 2018A Statistical Recurrent Model on the Manifold of Symmetric Positive Definite Matricescited by 18
  8. 2020Generalization bound of globally optimal non-convex neural network training: Transportation map estimation by infinite dimensional Langevin dynamicscited by 14
  9. 2017Deep manifold-to-manifold transforming network for action recognitioncited by 9
  10. 2017Learning a Robust Representation via a Deep Network on Symmetric Positive Definite Manifoldscited by 7
  11. 2021Ghosts in Neural Networks: Existence, Structure and Role of Infinite-Dimensional Null Spacecited by 7
  12. 2020Banach Space Representer Theorems for Neural Networks and Ridge Splinescited by 3

Built on by

Cited by year (2026 to 7 Oct)

Citing papers per year: 2022: 1; 2023: 6; 2024: 3; 2025: 2; 2026: 2.

14 papers.

Sort:

  1. 2024Motif-aware Riemannian Graph Neural Network with Generative-Contrastive Learningcited by 23
  2. 2023Riemannian Residual Neural Networkscited by 21
  3. 2022Universality of Group Convolutional Neural Networks Based on Ridgelet Analysis on Groupscited by 11
  4. 2023Horospherical Decision Boundaries for Large Margin Classification in Hyperbolic Spacecited by 11
  5. 2023Hyperbolic Random Forestscited by 9
  6. 2023Quantum Ridgelet Transform: Winning Lottery Ticket of Neural Networks with Quantum Computationcited by 5
  7. 2023Joint Group Invariant Functions on Data-Parameter Domain Induce Universal Neural Networkscited by 5
  8. 2026Neural Networks on Symmetric Spaces of Noncompact Typecited by 5
  9. 2023Deep Ridgelet Transform: Voice with Koopman Operator Proves Universality of Formal Deep Networkscited by 2
  10. 2024A unified Fourier slice method to derive ridgelet transform for a variety of depth-2 neural networkscited by 2
  11. 2024Deep Ridgelet Transform and Unified Universality Theorem for Deep and Shallow Joint-Group-Equivariant Machinescited by 2
  12. 2025Busemann Functions in the Wasserstein Space: Existence, Closed-Forms, and Applications to Slicingcited by 1
2 more
  1. 2025Siegel Neural Networkscited by 1
  2. 20261-Lipschitz Neural Networks on Hadamard Manifoldscited by 0