# Learning the Geometry of Data: A Mathematical Review of Shape Space Analysis

arXiv: https://arxiv.org/abs/2606.17022 · HTML: https://mlsanity.com/paper/2606.17022/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 10 of its 535 references that are in the map.
- First posted on arXiv 15 Jun 2026.

## Builds on (most cited first, 10 of 10)

- [Adversarial Autoencoders](https://mlsanity.com/paper/1511.05644/) (2015)
- [Topological Deep Learning: Going Beyond Graph Data](https://mlsanity.com/paper/2206.00606/) (2022)
- [SOM-VAE: Interpretable Discrete Representation Learning on Time Series](https://mlsanity.com/paper/1806.02199/) (2018)
- [Robust Watertight Manifold Surface Generation Method for ShapeNet Models](https://mlsanity.com/paper/1802.01698/) (2018)
- [Deep Level Sets: Implicit Surface Representations for 3D Shape Inference](https://mlsanity.com/paper/1901.06802/) (2019)
- [Score matching for sub-Riemannian bridge sampling](https://mlsanity.com/paper/2404.15258/) (2024)
- [Self Supervised Networks for Learning Latent Space Representations of Human Body Scans and Motions](https://mlsanity.com/paper/2411.03475/) (2024)
- [GenAnalysis: Joint Shape Analysis by Learning Man-Made Shape Generators with Deformation Regularizations](https://mlsanity.com/paper/2503.00807/) (2025)
- [En masse scanning and automated surfacing of small objects using Micro-CT](https://mlsanity.com/paper/2410.07385/) (2024)
- [Transformed Multi-view 3D Shape Features with Contrastive Learning](https://mlsanity.com/paper/2510.19955/) (2025)

## Abstract (arXiv)

A central objective of machine learning is to identify structure and patterns in data. Advances in data acquisition have increasingly produced datasets whose observations possess rich geometric form, giving rise to shape spaces that encode variability in object geometry. Such datasets arise across a wide range of disciplines, including biology, medicine, anthropology, and computer vision, where subtle geometric differences often carry important scientific information. Traditional machine learning methods, however, are frequently ill-equipped to account for the nonlinear geometric structure underlying these data. This survey synthesizes a rapidly growing body of work on shape space analysis, which provides a mathematical and computational framework for the study of geometric data. Drawing on ideas from differential geometry, statistics, and machine learning, we organize the literature around a common analytical pipeline: shape representation and parameterization, the rigorous construction of robust geodesic metrics, statistical analysis on shape spaces, and geometry-aware learning methods. We discuss how these tools enable the characterization of shape variability, the comparison of geometric objects, and the analysis of structural trajectories across populations and time. To illustrate the breadth of the field, we highlight applications spanning multiple scales of biological organization, including studies of subcellular morphology and primate tooth evolution. Across these and many other domains, researchers face common challenges arising from complex, nonlinear, and often unaligned geometric variation. The review concludes by identifying key theoretical and computational challenges, as well as emerging opportunities driven by increasingly large and diverse geometric datasets.