# Product Manifold Learning

arXiv: https://arxiv.org/abs/2010.09908 · HTML: https://mlsanity.com/paper/2010.09908/

## Key facts

- Cited by 16 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 19 Oct 2020.

## Built on by (most cited first, 16 of 16)

- [CoNeRF: Controllable Neural Radiance Fields](https://mlsanity.com/paper/2112.01983/) (2021)
- [Neural Latent Geometry Search: Product Manifold Inference via Gromov-Hausdorff-Informed Bayesian Optimization](https://mlsanity.com/paper/2309.04810/) (2023)
- [LDLE: Low Distortion Local Eigenmaps](https://mlsanity.com/paper/2101.11055/) (2021)
- [Multi-modal Differentiable Unsupervised Feature Selection](https://mlsanity.com/paper/2303.09381/) (2023)
- [Learning Cartesian Product Graphs with Laplacian Constraints](https://mlsanity.com/paper/2402.08105/) (2024)
- [Enriching Disentanglement: From Logical Definitions to Quantitative Metrics](https://mlsanity.com/paper/2305.11512/) (2023)
- [Mixed-curvature decision trees and random forests](https://mlsanity.com/paper/2410.13879/) (2024)
- [Toward a Unified Geometry Understanding: Riemannian Diffusion Framework for Graph Generation and Prediction](https://mlsanity.com/paper/2510.04522/) (2025)
- [Mode and Ridge Estimation in Euclidean and Directional Product Spaces: A Mean Shift Approach](https://mlsanity.com/paper/2110.08505/) (2021)
- [Spectral Self-supervised Feature Selection](https://mlsanity.com/paper/2407.09061/) (2024)
- [In-Context Semi-Supervised Learning](https://mlsanity.com/paper/2512.15934/) (2025)
- [PDF-HR: Pose Distance Fields for Humanoid Robots](https://mlsanity.com/paper/2602.04851/) (2026)
- [Parametrizing Product Shape Manifolds by Composite Networks](https://mlsanity.com/paper/2302.14665/) (2023)
- [The G-invariant graph Laplacian](https://mlsanity.com/paper/2303.17001/) (2023)
- [Gromov-Hausdorff Distances for Comparing Product Manifolds of Model Spaces](https://mlsanity.com/paper/2309.05678/) (2023)
- [Mixed-Curvature Decision Trees and Random Forests](https://mlsanity.com/paper/2406.05227/) (2024)

## Abstract (arXiv)

We consider problems of dimensionality reduction and learning data representations for continuous spaces with two or more independent degrees of freedom. Such problems occur, for example, when observing shapes with several components that move independently. Mathematically, if the parameter space of each continuous independent motion is a manifold, then their combination is known as a product manifold. In this paper, we present a new paradigm for non-linear independent component analysis called manifold factorization. Our factorization algorithm is based on spectral graph methods for manifold learning and the separability of the Laplacian operator on product spaces. Recovering the factors of a manifold yields meaningful lower-dimensional representations and provides a new way to focus on particular aspects of the data space while ignoring others. We demonstrate the potential use of our method for an important and challenging problem in structural biology: mapping the motions of proteins and other large molecules using cryo-electron microscopy datasets.