A singular Riemannian geometry approach to Deep Neural Networks I. Theoretical foundations
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Summary
It is proved that the Kolmogorov quotient of this pseudometric space yields a smooth manifold, which is the base space of a particular vertical bundle.
- Type
- preprint
- Published
- 2021-12-17
- Cited by
- 15
- References
- 51
- Access
- Open access
- OpenAlex
- https://openalex.org/W4226030786
- Semantic Scholar
- https://api.semanticscholar.org/CorpusID:246240238
Keywords
Riemannian geometry, Mathematics, Manifold (fluid mechanics), Quotient, Riemannian manifold
References
- Deep Convolutional Networks on Graph-Structured Data
- Singular Semi-Riemannian Geometry
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- Perturbation theory for linear operators
- Lightlike Submanifolds of Semi-Riemannian Manifolds and Applications
- A comprehensive introduction to differential geometry
- Perturbation Theory of Eigenvalue Problems
- Variétés : différentielles et analytiques : fascicule de résultats
- Pseudoconnections and Manifolds with Degenerate Metrics
- Generalized connections in vector bundles
- Degenerate Riemannian metrics
- Approximation by superpositions of a sigmoidal function
- Multilayer feedforward networks are universal approximators
- Non-linear dimensionality reduction: Riemannian metric estimation and the problem of geometric discovery
- Learning shape correspondence with anisotropic convolutional neural networks
- Geometric Deep Learning on Graphs and Manifolds Using Mixture Model CNNs
- Geometric Deep Learning: Going beyond Euclidean data
- Introduction to the Theory of Lie Groups
Cited by
- Origami in N dimensions: How feed-forward networks manufacture linear separability
- Probabilistic Risk Assessment of an Obstacle Detection System for GoA 4 Freight Trains
- A singular Riemannian Geometry Approach to Deep Neural Networks III. Piecewise Differentiable Layers and Random Walks on n-dimensional Classes
- Deep learning as Ricci flow
- Unveiling Transformer Perception by Exploring Input Manifolds
- AI-Powered Approaches for Hypersurface Reconstruction in Multidimensional Spaces
- GeloVec: Higher Dimensional Geometric Smoothing for Coherent Visual Feature Extraction in Image Segmentation
- Emergent Riemannian geometry over learning discrete computations on continuous manifolds
- RNNs perform task computations by dynamically warping neural representations
- RiemannInfer: improving transformer inference through Riemannian geometry
- A singular Riemannian geometry approach to Deep Neural Networks II. Reconstruction of 1-D equivalence classes
- Prime Convolutional Model: Breaking the Ground for Theoretical Explainability
- A Geometric Flow Perspective on Continuous-Depth Networks
- Evaluating Interpretable Methods via Geometric Alignment of Functional Distortions
- GARFLN: Geodesic Adaptive Riemannian Functional Link Network
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