# Geometry-Aware Generative Autoencoders for Warped Riemannian Metric Learning and Generative Modeling on Data Manifolds

arXiv: https://arxiv.org/abs/2410.12779 · HTML: https://mlsanity.com/paper/2410.12779/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- Builds on 25 of its 54 references that are in the map.
- First posted on arXiv 16 Oct 2024.

## Builds on (most cited first, 25 of 25)

- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Generative Adversarial Networks](https://mlsanity.com/paper/2203.00667/) (2022)
- [Spectral Normalization for Generative Adversarial Networks](https://mlsanity.com/paper/1802.05957/) (2018)
- [Improving and generalizing flow-based generative models with minibatch optimal transport](https://mlsanity.com/paper/2302.00482/) (2023)
- [Flow Matching on General Geometries](https://mlsanity.com/paper/2302.03660/) (2023)
- [TrajectoryNet: A Dynamic Optimal Transport Network for Modeling Cellular Dynamics](https://mlsanity.com/paper/2002.04461/) (2020)
- [Diffusion Schr"odinger Bridge Matching](https://mlsanity.com/paper/2303.16852/) (2023)
- [Simulation-free Schr"odinger bridges via score and flow matching](https://mlsanity.com/paper/2307.03672/) (2023)
- [Metric Flow Matching for Smooth Interpolations on the Data Manifold](https://mlsanity.com/paper/2405.14780/) (2024)
- [Manifold Interpolating Optimal-Transport Flows for Trajectory Inference](https://mlsanity.com/paper/2206.14928/) (2022)
- [A Computational Framework for Solving Wasserstein Lagrangian Flows](https://mlsanity.com/paper/2310.10649/) (2023)
- [Extendable and invertible manifold learning with geometry regularized autoencoders](https://mlsanity.com/paper/2007.07142/) (2020)
- [ImageFlowNet: Forecasting Multiscale Image-Level Trajectories of Disease Progression with Irregularly-Sampled Longitudinal Medical Images](https://mlsanity.com/paper/2406.14794/) (2024)
- [Assessing Neural Network Representations During Training Using Noise-Resilient Diffusion Spectral Entropy](https://mlsanity.com/paper/2312.04823/) (2023)
- [Riemannian Diffusion Schr"odinger Bridge](https://mlsanity.com/paper/2207.03024/) (2022)
- [On Linear Interpolation in the Latent Space of Deep Generative Models](https://mlsanity.com/paper/2105.03663/) (2021)
- [Learning shared neural manifolds from multi-subject FMRI data](https://mlsanity.com/paper/2201.00622/) (2021)
- [CUTS: A Deep Learning and Topological Framework for Multigranular Unsupervised Medical Image Segmentation](https://mlsanity.com/paper/2209.11359/) (2022)
- [Revisiting Latent-Space Interpolation via a Quantitative Evaluation Framework](https://mlsanity.com/paper/2110.06421/) (2021)
- [A Heat Diffusion Perspective on Geodesic Preserving Dimensionality Reduction](https://mlsanity.com/paper/2305.19043/) (2023)
- [DiffKillR: Killing and Recreating Diffeomorphisms for Cell Annotation in Dense Microscopy Images](https://mlsanity.com/paper/2410.03058/) (2024)
- [Hyperedge Representations with Hypergraph Wavelets: Applications to Spatial Transcriptomics](https://mlsanity.com/paper/2409.09469/) (2024)
- [Neural FIM for learning Fisher Information Metrics from point cloud data](https://mlsanity.com/paper/2306.06062/) (2023)
- [DYMAG: Rethinking Message Passing Using Dynamical-systems-based Waveforms](https://mlsanity.com/paper/2309.09924/) (2023)
- [Monotone Generative Modeling via a Gromov-Monge Embedding](https://mlsanity.com/paper/2311.01375/) (2023)

## Built on by (most cited first, 13 of 13)

- [Staying VIGILant: Mitigating Visual Laziness via Counterfactual Visual Alignment in MLLMs](https://mlsanity.com/paper/2606.26387/) (2026)
- [Follow the Energy, Find the Path: Riemannian Metrics from Energy-Based Models](https://mlsanity.com/paper/2505.18230/) (2025)
- [Enforcing Latent Euclidean Geometry in Single-Cell VAEs for Manifold Interpolation](https://mlsanity.com/paper/2507.11789/) (2025)
- [Riemannian Generative Decoder](https://mlsanity.com/paper/2506.19133/) (2025)
- [Geodesic Calculus on Implicitly Defined Latent Manifolds](https://mlsanity.com/paper/2510.09468/) (2025)
- [Dispersion Loss Counteracts Embedding Condensation and Improves Generalization in Small Language Models](https://mlsanity.com/paper/2602.00217/) (2026)
- [Riemannian-Manifold Steering: Geometry-Aware Generative Autoencoders for Label-Free Steering](https://mlsanity.com/paper/2605.24942/) (2026)
- [ProtScape: A molecular structure and energy-aware representation for protein conformation generation](https://mlsanity.com/paper/2410.20317/) (2024)
- [Geometric Embedding Alignment via Curvature Matching in Transfer Learning](https://mlsanity.com/paper/2506.13015/) (2025)
- [Energy Guided Geometric Flow Matching](https://mlsanity.com/paper/2509.25230/) (2025)
- [Multi-marginal temporal Schr"odinger Bridge Matching from unpaired data](https://mlsanity.com/paper/2510.01894/) (2025)
- [RNAGenScape: Property-Guided, Optimized Generation of mRNA Sequences with Manifold Langevin Dynamics](https://mlsanity.com/paper/2510.24736/) (2025)
- [A Non-Adversarial Approach to Idempotent Generative Modelling](https://mlsanity.com/paper/2511.02614/) (2025)

## Abstract (arXiv)

Rapid growth of high-dimensional datasets in fields such as single-cell RNA sequencing and spatial genomics has led to unprecedented opportunities for scientific discovery, but it also presents unique computational and statistical challenges. Traditional methods struggle with geometry-aware data generation, interpolation along meaningful trajectories, and transporting populations via feasible paths. To address these issues, we introduce Geometry-Aware Generative Autoencoder (GAGA), a novel framework that combines extensible manifold learning with generative modeling. GAGA constructs a neural network embedding space that respects the intrinsic geometries discovered by manifold learning and learns a novel warped Riemannian metric on the data space. This warped metric is derived from both the points on the data manifold and negative samples off the manifold, allowing it to characterize a meaningful geometry across the entire latent space. Using this metric, GAGA can uniformly sample points on the manifold, generate points along geodesics, and interpolate between populations across the learned manifold using geodesic-guided flows. GAGA shows competitive performance in simulated and real-world datasets, including a 30% improvement over the state-of-the-art methods in single-cell population-level trajectory inference.