# A singular Riemannian geometry approach to Deep Neural Networks II. Reconstruction of 1-D equivalence classes

arXiv: https://arxiv.org/abs/2112.10583 · HTML: https://mlsanity.com/paper/2112.10583/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 34 references that are in the map.
- First posted on arXiv 17 Dec 2021.

## Builds on (most cited first, 13 of 13)

- [Implicit Neural Representations with Periodic Activation Functions](https://mlsanity.com/paper/2006.09661/) (2020)
- [Geometric deep learning: going beyond Euclidean data](https://mlsanity.com/paper/1611.08097/) (2016)
- [Generative Adversarial Networks: An Overview](https://mlsanity.com/paper/1710.07035/) (2017)
- [Geometric deep learning on graphs and manifolds using mixture model CNNs](https://mlsanity.com/paper/1611.08402/) (2016)
- [Mitigating Adversarial Effects Through Randomization](https://mlsanity.com/paper/1711.01991/) (2017)
- [Audio Adversarial Examples: Targeted Attacks on Speech-to-Text](https://mlsanity.com/paper/1801.01944/) (2018)
- [Deep Convolutional Networks on Graph-Structured Data](https://mlsanity.com/paper/1506.05163/) (2015)
- [MultiResUNet : Rethinking the U-Net Architecture for Multimodal Biomedical Image Segmentation](https://mlsanity.com/paper/1902.04049/) (2019)
- [The loss surface of deep and wide neural networks](https://mlsanity.com/paper/1704.08045/) (2017)
- [Boundary-weighted Domain Adaptive Neural Network for Prostate MR Image Segmentation](https://mlsanity.com/paper/1902.08128/) (2019)
- [Deep neural network enabled corrective source term approach to hybrid analysis and modeling](https://mlsanity.com/paper/2105.11521/) (2021)
- [Imitation and Mirror Systems in Robots through Deep Modality Blending Networks](https://mlsanity.com/paper/2106.08422/) (2021)
- [A Differential Topological View of Challenges in Learning with Feedforward Neural Networks](https://mlsanity.com/paper/1811.10304/) (2018)

## Built on by (most cited first, 8 of 8)

- [Origami in N dimensions: How feed-forward networks manufacture linear separability](https://mlsanity.com/paper/2203.11355/) (2022)
- [Probabilistic Risk Assessment of an Obstacle Detection System for GoA 4 Freight Trains](https://mlsanity.com/paper/2306.14814/) (2023)
- [A singular Riemannian Geometry Approach to Deep Neural Networks III. Piecewise Differentiable Layers and Random Walks on n-dimensional Classes](https://mlsanity.com/paper/2404.06104/) (2024)
- [GeloVec: Higher Dimensional Geometric Smoothing for Coherent Visual Feature Extraction in Image Segmentation](https://mlsanity.com/paper/2505.01057/) (2025)
- [Unveiling Transformer Perception by Exploring Input Manifolds](https://mlsanity.com/paper/2410.06019/) (2024)
- [Prime Convolutional Model: Breaking the Ground for Theoretical Explainability](https://mlsanity.com/paper/2503.02773/) (2025)
- [Riemannian Optimization in Modular Systems](https://mlsanity.com/paper/2603.03610/) (2026)
- [Affine Geometry of Gaussian ReLU Networks via Conditional Kac-Rice Formulas](https://mlsanity.com/paper/2609.32695/) (2026)

## Abstract (arXiv)

In a previous work, we proposed a geometric framework to study a deep neural network, seen as sequence of maps between manifolds, employing singular Riemannian geometry. In this paper, we present an application of this framework, proposing a way to build the class of equivalence of an input point: such class is defined as the set of the points on the input manifold mapped to the same output by the neural network. In other words, we build the preimage of a point in the output manifold in the input space. In particular. we focus for simplicity on the case of neural networks maps from n-dimensional real spaces to (n - 1)-dimensional real spaces, we propose an algorithm allowing to build the set of points lying on the same class of equivalence. This approach leads to two main applications: the generation of new synthetic data and it may provides some insights on how a classifier can be confused by small perturbation on the input data (e.g. a penguin image classified as an image containing a chihuahua). In addition, for neural networks from 2D to 1D real spaces, we also discuss how to find the preimages of closed intervals of the real line. We also present some numerical experiments with several neural networks trained to perform non-linear regression tasks, including the case of a binary classifier.