# Gamma-VAE: Curvature regularized variational autoencoders for uncovering emergent low dimensional geometric structure in high dimensional data

arXiv: https://arxiv.org/abs/2403.01078 · HTML: https://mlsanity.com/paper/2403.01078/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 42 references that are in the map.
- First posted on arXiv 2 Mar 2024.

## Builds on (most cited first, 12 of 12)

- [UMAP: Uniform Manifold Approximation and Projection for Dimension Reduction](https://mlsanity.com/paper/1802.03426/) (2018)
- [Neural Tangent Kernel: Convergence and Generalization in Neural Networks](https://mlsanity.com/paper/1806.07572/) (2018)
- [An Introduction to Variational Autoencoders](https://mlsanity.com/paper/1906.02691/) (2019)
- [Representation Learning on Graphs: Methods and Applications](https://mlsanity.com/paper/1709.05584/) (2017)
- [Hyperspherical Variational Auto-Encoders](https://mlsanity.com/paper/1804.00891/) (2018)
- [Latent Space Oddity: on the Curvature of Deep Generative Models](https://mlsanity.com/paper/1710.11379/) (2017)
- [The Recurrent Neural Tangent Kernel](https://mlsanity.com/paper/2006.10246/) (2020)
- [A Sober Look at the Unsupervised Learning of Disentangled Representations and their Evaluation](https://mlsanity.com/paper/2010.14766/) (2020)
- [Latent Variable Modelling with Hyperbolic Normalizing Flows](https://mlsanity.com/paper/2002.06336/) (2020)
- [On Explicit Curvature Regularization in Deep Generative Models](https://mlsanity.com/paper/2309.10237/) (2023)
- [Beta-VAE Reproducibility: Challenges and Extensions](https://mlsanity.com/paper/2112.14278/) (2021)
- [Quantifying Extrinsic Curvature in Neural Manifolds](https://mlsanity.com/paper/2212.10414/) (2022)

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

- [On Geometry Regularization in Autoencoder Reduced-Order Models with Latent Neural ODE Dynamics](https://mlsanity.com/paper/2603.03238/) (2026)
- [Thinner Latent Spaces: Detecting Dimension and Imposing Invariance with Conformal Autoencoders](https://mlsanity.com/paper/2408.16138/) (2024)

## Abstract (arXiv)

Natural systems with emergent behaviors often organize along low-dimensional subsets of high-dimensional spaces. For example, despite the tens of thousands of genes in the human genome, the principled study of genomics is fruitful because biological processes rely on coordinated organization that results in lower dimensional phenotypes. To uncover this organization, many nonlinear dimensionality reduction techniques have successfully embedded high-dimensional data into low-dimensional spaces by preserving local similarities between data points. However, the nonlinearities in these methods allow for too much curvature to preserve general trends across multiple non-neighboring data clusters, thereby limiting their interpretability and generalizability to out-of-distribution data. Here, we address both of these limitations by regularizing the curvature of manifolds generated by variational autoencoders, a process we coin ``Gamma-VAE''. We demonstrate its utility using two example data sets: bulk RNA-seq from the The Cancer Genome Atlas (TCGA) and the Genotype Tissue Expression (GTEx); and single cell RNA-seq from a lineage tracing experiment in hematopoietic stem cell differentiation. We find that the resulting regularized manifolds identify mesoscale structure associated with different cancer cell types, and accurately re-embed tissues from completely unseen, out-of distribution cancers as if they were originally trained on them. Finally, we show that preserving long-range relationships to differentiated cells separates undifferentiated cells -- which have not yet specialized -- according to their eventual fate. Broadly, we anticipate that regularizing the curvature of generative models will enable more consistent, predictive, and generalizable models in any high-dimensional system with emergent low-dimensional behavior.