# Adversarial Autoencoders with Constant-Curvature Latent Manifolds

arXiv: https://arxiv.org/abs/1812.04314 · HTML: https://mlsanity.com/paper/1812.04314/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 11 Dec 2018.

## Built on by (most cited first, 12 of 12)

- [Hyperbolic Deep Neural Networks: A Survey](https://mlsanity.com/paper/2101.04562/) (2021)
- [Hyperbolic Geometric Latent Diffusion Model for Graph Generation](https://mlsanity.com/paper/2405.03188/) (2024)
- [Unsupervised Hyperbolic Representation Learning via Message Passing Auto-Encoders](https://mlsanity.com/paper/2103.16046/) (2021)
- [Curvature Graph Generative Adversarial Networks](https://mlsanity.com/paper/2203.01604/) (2022)
- [Wrapped Distributions on homogeneous Riemannian manifolds](https://mlsanity.com/paper/2204.09790/) (2022)
- [Atlas Based Representation and Metric Learning on Manifolds](https://mlsanity.com/paper/2106.07062/) (2021)
- [State Representation Learning Using an Unbalanced Atlas](https://mlsanity.com/paper/2305.10267/) (2023)
- [GGBall: Graph Generative Model on Poincar'e Ball](https://mlsanity.com/paper/2506.07198/) (2025)
- [Riemannian Generative Decoder](https://mlsanity.com/paper/2506.19133/) (2025)
- [Variational autoencoders with latent high-dimensional steady geometric flows for dynamics](https://mlsanity.com/paper/2410.10137/) (2024)
- [Geometric Embedding Alignment via Curvature Matching in Transfer Learning](https://mlsanity.com/paper/2506.13015/) (2025)
- [Hyperbolic Graph Embeddings: a Survey and an Evaluation on Anomaly Detection](https://mlsanity.com/paper/2512.18826/) (2025)

## Abstract (arXiv)

Constant-curvature Riemannian manifolds (CCMs) have been shown to be ideal embedding spaces in many application domains, as their non-Euclidean geometry can naturally account for some relevant properties of data, like hierarchy and circularity. In this work, we introduce the CCM adversarial autoencoder (CCM-AAE), a probabilistic generative model trained to represent a data distribution on a CCM. Our method works by matching the aggregated posterior of the CCM-AAE with a probability distribution defined on a CCM, so that the encoder implicitly learns to represent data on the CCM to fool the discriminator network. The geometric constraint is also explicitly imposed by jointly training the CCM-AAE to maximise the membership degree of the embeddings to the CCM. While a few works in recent literature make use of either hyperspherical or hyperbolic manifolds for different learning tasks, ours is the first unified framework to seamlessly deal with CCMs of different curvatures. We show the effectiveness of our model on three different datasets characterised by non-trivial geometry: semi-supervised classification on MNIST, link prediction on two popular citation datasets, and graph-based molecule generation using the QM9 chemical database. Results show that our method improves upon other autoencoders based on Euclidean and non-Euclidean geometries on all tasks taken into account.