mlsanityData to 7 Oct 2026

A singular Riemannian geometry approach to Deep Neural Networks I. Theoretical foundations

arXiv 2201.09656 ↗First posted 17 Dec 2021cs.LG cs.NE math.MGcited by 11

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Abstract (arXiv)

Deep Neural Networks are widely used for solving complex problems in several scientific areas, such as speech recognition, machine translation, image analysis. The strategies employed to investigate their theoretical properties mainly rely on Euclidean geometry, but in the last years new approaches based on Riemannian geometry have been developed. Motivated by some open problems, we study a particular sequence of maps between manifolds, with the last manifold of the sequence equipped with a Riemannian metric. We investigate the structures induced trough pullbacks on the other manifolds of the sequence and on some related quotients. In particular, we show that the pullbacks of the final Riemannian metric to any manifolds of the sequence is a degenerate Riemannian metric inducing a structure of pseudometric space, we show that the Kolmogorov quotient of this pseudometric space yields a smooth manifold, which is the base space of a particular vertical bundle. We investigate the theoretical properties of the maps of such sequence, eventually we focus on the case of maps between manifolds implementing neural networks of practical interest and we present some applications of the geometric framework we introduced in the first part of the paper.

Key facts. Cited by 11 papers in the map (data through 7 Oct 2026). Builds on 16 of its 51 references that are in the map. First posted on arXiv 17 Dec 2021.

Builds on

16 of 51 references.

References · 16

  1. 2016Geometric deep learning: going beyond Euclidean datacited by 1,042
  2. 2016Geometric deep learning on graphs and manifolds using mixture model CNNscited by 468
  3. 2015Deep Convolutional Networks on Graph-Structured Datacited by 246
  4. 2019Universal Approximation with Deep Narrow Networkscited by 198
  5. 2018An overview of deep learning in medical imaging focusing on MRIcited by 140
  6. 2017Universal Function Approximation by Deep Neural Nets with Bounded Width and ReLU Activationscited by 104
  7. 2016Learning shape correspondence with anisotropic convolutional neural networkscited by 95
  8. 2020A Survey of Deep Learning Techniques for Neural Machine Translationcited by 50
  9. 2020Non-Euclidean Universal Approximationcited by 29
  10. 2018A survey on Deep Learning Advances on Different 3D Data Representationscited by 25
  11. 2019Community detection in node-attributed social networks: a surveycited by 24
  12. 2013Non-linear dimensionality reduction: Riemannian metric estimation and the problem of geometric discoverycited by 14
4 more
  1. 2020Deep Neural Networks for Automatic Speech Processing: A Survey from Large Corpora to Limited Datacited by 9
  2. 2021Multi-view Data Visualisation via Manifold Learningcited by 7
  3. 2020Rehabilitating Isomap: Euclidean Representation of Geodesic Structurecited by 5
  4. 2018A Differential Topological View of Challenges in Learning with Feedforward Neural Networkscited by 4

Built on by

Cited by year (2026 to 7 Oct)

Citing papers per year: 2022: 1; 2023: 1; 2024: 3; 2025: 4; 2026: 2.

11 papers.

  1. 2022Origami in N dimensions: How feed-forward networks manufacture linear separabilitycited by 9
  2. 2024Deep Learning as Ricci Flowcited by 4
  3. 2023Probabilistic Risk Assessment of an Obstacle Detection System for GoA 4 Freight Trainscited by 3
  4. 2025Emergent Riemannian geometry over learning discrete computations on continuous manifoldscited by 3
  5. 2024A singular Riemannian Geometry Approach to Deep Neural Networks III. Piecewise Differentiable Layers and Random Walks on n-dimensional Classescited by 2
  6. 2025GeloVec: Higher Dimensional Geometric Smoothing for Coherent Visual Feature Extraction in Image Segmentationcited by 2
  7. 2025RNNs perform task computations by dynamically warping neural representationscited by 1
  8. 2024Unveiling Transformer Perception by Exploring Input Manifoldscited by 0
  9. 2025Prime Convolutional Model: Breaking the Ground for Theoretical Explainabilitycited by 0
  10. 2026Dual-Manifold Geometry Guided Representation Learning: Adaptive Coupling between Kernel and Data Spacescited by 0
  11. 2026Task-Induced Riemannian Metrics for Vision Transformer Feature Spacescited by 0