# Thinner Latent Spaces: Detecting Dimension and Imposing Invariance with Conformal Autoencoders

arXiv: https://arxiv.org/abs/2408.16138 · HTML: https://mlsanity.com/paper/2408.16138/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 15 of its 51 references that are in the map.
- First posted on arXiv 28 Aug 2024.

## Builds on (most cited first, 15 of 15)

- [Efficient and Accurate Estimation of Lipschitz Constants for Deep Neural Networks](https://mlsanity.com/paper/1906.04893/) (2019)
- [Sobolev Training for Neural Networks](https://mlsanity.com/paper/1706.04859/) (2017)
- [Equivariance with Learned Canonicalization Functions](https://mlsanity.com/paper/2211.06489/) (2022)
- [Training robust neural networks using Lipschitz bounds](https://mlsanity.com/paper/2005.02929/) (2020)
- [Variational Autoencoders Pursue PCA Directions (by Accident)](https://mlsanity.com/paper/1812.06775/) (2018)
- [On Implicit Regularization in β-VAEs](https://mlsanity.com/paper/2002.00041/) (2020)
- [Chart Auto-Encoders for Manifold Structured Data](https://mlsanity.com/paper/1912.10094/) (2019)
- [Data-Driven Reduced-Order Modeling of Spatiotemporal Chaos with Neural Ordinary Differential Equations](https://mlsanity.com/paper/2109.00060/) (2021)
- [Improved Surrogates in Inertial Confinement Fusion with Manifold and Cycle Consistencies](https://mlsanity.com/paper/1912.08113/) (2019)
- [On the Parameter Combinations That Matter and on Those That do Not](https://mlsanity.com/paper/2110.06717/) (2021)
- [Group invariant machine learning by fundamental domain projections](https://mlsanity.com/paper/2202.02164/) (2022)
- [Nonlinear dimensionality reduction then and now: AIMs for dissipative PDEs in the ML era](https://mlsanity.com/paper/2310.15816/) (2023)
- [Toroidal Coordinates: Decorrelating Circular Coordinates With Lattice Reduction](https://mlsanity.com/paper/2212.07201/) (2022)
- [Gamma-VAE: Curvature regularized variational autoencoders for uncovering emergent low dimensional geometric structure in high dimensional data](https://mlsanity.com/paper/2403.01078/) (2024)
- [Gentlest ascent dynamics on manifolds defined by adaptively sampled point-clouds](https://mlsanity.com/paper/2302.04426/) (2023)

## Abstract (arXiv)

Conformal Autoencoders are a neural network architecture that imposes orthogonality conditions between the gradients of latent variables to obtain disentangled representations of data. In this work we show that orthogonality relations within the latent layer of the network can be leveraged to infer the intrinsic dimensionality of nonlinear manifold data sets (locally characterized by the dimension of their tangent space), while simultaneously computing encoding and decoding (embedding) maps. We outline the relevant theory relying on differential geometry, and describe the corresponding gradient-descent optimization algorithm. The method is applied to several data sets and we highlight its applicability, advantages, and shortcomings. In addition, we demonstrate that the same computational technology can be used to build coordinate invariance to local group actions when defined only on a (reduced) submanifold of the embedding space.