# Discovering Data Manifold Geometry through Geometric Properties

arXiv: https://arxiv.org/abs/2602.02611 · HTML: https://mlsanity.com/paper/2602.02611/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 32 of its 44 references that are in the map.
- First posted on arXiv 2 Feb 2026.

## Builds on (most cited first, 30 of 32)

- [A Simple Framework for Contrastive Learning of Visual Representations](https://mlsanity.com/paper/2002.05709/) (2020)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Momentum Contrast for Unsupervised Visual Representation Learning](https://mlsanity.com/paper/1911.05722/) (2019)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Barlow Twins: Self-Supervised Learning via Redundancy Reduction](https://mlsanity.com/paper/2103.03230/) (2021)
- [Concept Bottleneck Models](https://mlsanity.com/paper/2007.04612/) (2020)
- [Challenging Common Assumptions in the Unsupervised Learning of Disentangled Representations](https://mlsanity.com/paper/1811.12359/) (2018)
- [Interpreting the Latent Space of GANs for Semantic Face Editing](https://mlsanity.com/paper/1907.10786/) (2019)
- [Disentangling by Factorising](https://mlsanity.com/paper/1802.05983/) (2018)
- [Neural Spline Flows](https://mlsanity.com/paper/1906.04032/) (2019)
- [Adversarial Feature Learning](https://mlsanity.com/paper/1605.09782/) (2016)
- [Towards a Definition of Disentangled Representations](https://mlsanity.com/paper/1812.02230/) (2018)
- [Residual Flows for Invertible Generative Modeling](https://mlsanity.com/paper/1906.02735/) (2019)
- [Adversarially Learned Inference](https://mlsanity.com/paper/1606.00704/) (2016)
- [Beyond Data and Model Parallelism for Deep Neural Networks](https://mlsanity.com/paper/1807.05358/) (2018)
- [CRAFT: Concept Recursive Activation FacTorization for Explainability](https://mlsanity.com/paper/2211.10154/) (2022)
- [Neural Lyapunov Control](https://mlsanity.com/paper/2005.00611/) (2020)
- [Equivariance with Learned Canonicalization Functions](https://mlsanity.com/paper/2211.06489/) (2022)
- [The Riemannian Geometry of Deep Generative Models](https://mlsanity.com/paper/1711.08014/) (2017)
- [Metric Flow Matching for Smooth Interpolations on the Data Manifold](https://mlsanity.com/paper/2405.14780/) (2024)
- [CaDeX: Learning Canonical Deformation Coordinate Space for Dynamic Surface Representation via Neural Homeomorphism](https://mlsanity.com/paper/2203.16529/) (2022)
- [LyaNet: A Lyapunov Framework for Training Neural ODEs](https://mlsanity.com/paper/2202.02526/) (2022)
- [Commutative Lie Group VAE for Disentanglement Learning](https://mlsanity.com/paper/2106.03375/) (2021)
- [Isometric Representation Learning for Disentangled Latent Space of Diffusion Models](https://mlsanity.com/paper/2407.11451/) (2024)
- [A Stable Whitening Optimizer for Efficient Neural Network Training](https://mlsanity.com/paper/2506.07254/) (2025)
- [Autoencoders for discovering manifold dimension and coordinates in data from complex dynamical systems](https://mlsanity.com/paper/2305.01090/) (2023)
- [Pullback Flow Matching on Data Manifolds](https://mlsanity.com/paper/2410.04543/) (2024)
- [Bundle Networks: Fiber Bundles, Local Trivializations, and a Generative Approach to Exploring Many-to-one Maps](https://mlsanity.com/paper/2110.06983/) (2021)
- [Follow the Energy, Find the Path: Riemannian Metrics from Energy-Based Models](https://mlsanity.com/paper/2505.18230/) (2025)
- [Robust Weight Initialization for Tanh Neural Networks with Fixed Point Analysis](https://mlsanity.com/paper/2410.02242/) (2024)

## Built on by (most cited first, 1 of 1)

- [Riemannian Archetypal Analysis: Interpretable non-linear data analysis on deformed star distributions](https://mlsanity.com/paper/2605.24113/) (2026)

## Abstract (arXiv)

A prevailing paradigm in modern representation learning is the map-first approach, in which a representation map is learned from reconstruction, embedding, or task objectives. At the optimum, when the learned map accurately recovers a global coordinate chart, it should exhibit three structural properties whose geometric meaning can be illustrated through a face-editing example: Commutativity requires that changing pose and then expression gives the same result as applying them in the reverse order; Time Coherence requires that the same variation along one coordinate induces the same expression change across faces; Common-Reference requires that all faces are organized relative to a common reference face. However, small approximation errors in the learned map need not translate into small errors in these structural properties, and can therefore disrupt the global organization of the representation. Based on this observation, we consider the converse of the map-first formulation and ask whether a global representation can instead emerge by directly learning these properties. We represent variations along individual coordinates through vector fields defined in the ambient space and introduce a non-contraction condition preventing one transformation from destroying directions associated with the others. We derive an unsupervised objective that learns these structural properties and establish theoretical results connecting its minimization to tangent-space recovery. Experiments on controlled manifolds validate the predicted tangent-space recovery and global structure, while an autoencoder baseline shows that small map-first errors can still produce substantial violations of the targeted properties.