# Representing Closed Transformation Paths in Encoded Network Latent Space

arXiv: https://arxiv.org/abs/1912.02644 · HTML: https://mlsanity.com/paper/1912.02644/

## Key facts

- Cited by 17 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 5 Dec 2019.

## Built on by (most cited first, 17 of 17)

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- [Hamiltonian latent operators for content and motion disentanglement in image sequences](https://mlsanity.com/paper/2112.01641/) (2021)
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- [Robust Self-Supervised Learning with Lie Groups](https://mlsanity.com/paper/2210.13356/) (2022)
- [Surrogate Modeling of 3D Rayleigh-Benard Convection with Equivariant Autoencoders](https://mlsanity.com/paper/2505.13569/) (2025)
- [Learning Internal Representations of 3D Transformations from 2D Projected Inputs](https://mlsanity.com/paper/2303.17776/) (2023)
- [Manifold Contrastive Learning with Variational Lie Group Operators](https://mlsanity.com/paper/2306.13544/) (2023)
- [MANGO: Learning Disentangled Image Transformation Manifolds with Grouped Operators](https://mlsanity.com/paper/2409.09542/) (2024)
- [Latent Equivariant Operators for Robust Object Recognition: Promises and Challenges](https://mlsanity.com/paper/2602.18406/) (2026)

## Abstract (arXiv)

Deep generative networks have been widely used for learning mappings from a low-dimensional latent space to a high-dimensional data space. In many cases, data transformations are defined by linear paths in this latent space. However, the Euclidean structure of the latent space may be a poor match for the underlying latent structure in the data. In this work, we incorporate a generative manifold model into the latent space of an autoencoder in order to learn the low-dimensional manifold structure from the data and adapt the latent space to accommodate this structure. In particular, we focus on applications in which the data has closed transformation paths which extend from a starting point and return to nearly the same point. Through experiments on data with natural closed transformation paths, we show that this model introduces the ability to learn the latent dynamics of complex systems, generate transformation paths, and classify samples that belong on the same transformation path.