# Enforcing Latent Euclidean Geometry in Single-Cell VAEs for Manifold Interpolation

arXiv: https://arxiv.org/abs/2507.11789 · HTML: https://mlsanity.com/paper/2507.11789/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 59 references that are in the map.
- First posted on arXiv 15 Jul 2025.

## Builds on (most cited first, 19 of 19)

- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Building Normalizing Flows with Stochastic Interpolants](https://mlsanity.com/paper/2209.15571/) (2022)
- [Improving and generalizing flow-based generative models with minibatch optimal transport](https://mlsanity.com/paper/2302.00482/) (2023)
- [Multisample Flow Matching: Straightening Flows with Minibatch Couplings](https://mlsanity.com/paper/2304.14772/) (2023)
- [TrajectoryNet: A Dynamic Optimal Transport Network for Modeling Cellular Dynamics](https://mlsanity.com/paper/2002.04461/) (2020)
- [Action Matching: Learning Stochastic Dynamics from Samples](https://mlsanity.com/paper/2210.06662/) (2022)
- [Proximal Optimal Transport Modeling of Population Dynamics](https://mlsanity.com/paper/2106.06345/) (2021)
- [Metric Flow Matching for Smooth Interpolations on the Data Manifold](https://mlsanity.com/paper/2405.14780/) (2024)
- [An elementary introduction to information geometry](https://mlsanity.com/paper/1808.08271/) (2018)
- [Manifold Interpolating Optimal-Transport Flows for Trajectory Inference](https://mlsanity.com/paper/2206.14928/) (2022)
- [Geometrically Enriched Latent Spaces](https://mlsanity.com/paper/2008.00565/) (2020)
- [Predicting Cellular Responses to Novel Drug Perturbations at a Single-Cell Resolution](https://mlsanity.com/paper/2204.13545/) (2022)
- [Learning Flat Latent Manifolds with VAEs](https://mlsanity.com/paper/2002.04881/) (2020)
- [Pulling back information geometry](https://mlsanity.com/paper/2106.05367/) (2021)
- [Geometric Autoencoders -- What You See is What You Decode](https://mlsanity.com/paper/2306.17638/) (2023)
- [Extendable and invertible manifold learning with geometry regularized autoencoders](https://mlsanity.com/paper/2007.07142/) (2020)
- [Geometry-Aware Generative Autoencoders for Warped Riemannian Metric Learning and Generative Modeling on Data Manifolds](https://mlsanity.com/paper/2410.12779/) (2024)
- [A Heat Diffusion Perspective on Geodesic Preserving Dimensionality Reduction](https://mlsanity.com/paper/2305.19043/) (2023)
- [Neural FIM for learning Fisher Information Metrics from point cloud data](https://mlsanity.com/paper/2306.06062/) (2023)

## Built on by (most cited first, 5 of 5)

- [Making Reconstruction FID Predictive of Diffusion Generation FID](https://mlsanity.com/paper/2603.05630/) (2026)
- [Learning Lineage-guided Geodesics with Finsler Geometry](https://mlsanity.com/paper/2603.16708/) (2026)
- [Energy Guided Geometric Flow Matching](https://mlsanity.com/paper/2509.25230/) (2025)
- [Complex variational autoencoders admit K"ahler structure](https://mlsanity.com/paper/2511.15172/) (2025)
- [Understanding Latent Diffusability via Fisher Geometry](https://mlsanity.com/paper/2604.02751/) (2026)

## Abstract (arXiv)

Latent space interpolations are a powerful tool for navigating deep generative models in applied settings. An example is single-cell RNA sequencing, where existing methods model cellular state transitions as latent space interpolations with variational autoencoders, often assuming linear shifts and Euclidean geometry. However, unless explicitly enforced, linear interpolations in the latent space may not correspond to geodesic paths on the data manifold, limiting methods that assume Euclidean geometry in the data representations. We introduce FlatVI, a novel training framework that regularises the latent manifold of discrete-likelihood variational autoencoders towards Euclidean geometry, specifically tailored for modelling single-cell count data. By encouraging straight lines in the latent space to approximate geodesic interpolations on the decoded single-cell manifold, FlatVI enhances compatibility with downstream approaches that assume Euclidean latent geometry. Experiments on synthetic data support the theoretical soundness of our approach, while applications to time-resolved single-cell RNA sequencing data demonstrate improved trajectory reconstruction and manifold interpolation.