# Deep neural networks on diffeomorphism groups for optimal shape reparameterization

arXiv: https://arxiv.org/abs/2207.11141 · HTML: https://mlsanity.com/paper/2207.11141/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 40 references that are in the map.
- First posted on arXiv 22 Jul 2022.

## Builds on (most cited first, 10 of 10)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Deep Image Prior](https://mlsanity.com/paper/1711.10925/) (2017)
- [Invertible Residual Networks](https://mlsanity.com/paper/1811.00995/) (2018)
- [Stable Architectures for Deep Neural Networks](https://mlsanity.com/paper/1705.03341/) (2017)
- [Understanding and Mitigating Exploding Inverses in Invertible Neural Networks](https://mlsanity.com/paper/2006.09347/) (2020)
- [Deep learning as optimal control problems: models and numerical methods](https://mlsanity.com/paper/1904.05657/) (2019)
- [Structure preserving deep learning](https://mlsanity.com/paper/2006.03364/) (2020)
- [Shape Analysis of Surfaces Using General Elastic Metrics](https://mlsanity.com/paper/1910.02045/) (2019)
- [Shape Analysis on Lie Groups with Applications in Computer Animation](https://mlsanity.com/paper/1506.00783/) (2015)

## Built on by (most cited first, 2 of 2)

- [Designing Stable Neural Networks using Convex Analysis and ODEs](https://mlsanity.com/paper/2306.17332/) (2023)
- [Personalized Convolutional Dictionary Learning of Physiological Time Series](https://mlsanity.com/paper/2503.07687/) (2025)

## Abstract (arXiv)

One of the fundamental problems in shape analysis is to align curves or surfaces before computing geodesic distances between their shapes. Finding the optimal reparametrization realizing this alignment is a computationally demanding task, typically done by solving an optimization problem on the diffeomorphism group. In this paper, we propose an algorithm for constructing approximations of orientation-preserving diffeomorphisms by composition of elementary diffeomorphisms. The algorithm is implemented using PyTorch, and is applicable for both unparametrized curves and surfaces. Moreover, we show universal approximation properties for the constructed architectures, and obtain bounds for the Lipschitz constants of the resulting diffeomorphisms.