# Learning disentangled representations via product manifold projection

arXiv: https://arxiv.org/abs/2103.01638 · HTML: https://mlsanity.com/paper/2103.01638/

## Key facts

- Cited by 20 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 46 references that are in the map.
- First posted on arXiv 2 Mar 2021.

## Builds on (most cited first, 20 of 20)

- [PointNet: Deep Learning on Point Sets for 3D Classification and Segmentation](https://mlsanity.com/paper/1612.00593/) (2016)
- [UMAP: Uniform Manifold Approximation and Projection for Dimension Reduction](https://mlsanity.com/paper/1802.03426/) (2018)
- [Challenging Common Assumptions in the Unsupervised Learning of Disentangled Representations](https://mlsanity.com/paper/1811.12359/) (2018)
- [Disentangling by Factorising](https://mlsanity.com/paper/1802.05983/) (2018)
- [Towards a Definition of Disentangled Representations](https://mlsanity.com/paper/1812.02230/) (2018)
- [Nonlinear ICA Using Auxiliary Variables and Generalized Contrastive Learning](https://mlsanity.com/paper/1805.08651/) (2018)
- [Unsupervised Feature Extraction by Time-Contrastive Learning and Nonlinear ICA](https://mlsanity.com/paper/1605.06336/) (2016)
- [Learning Blind Video Temporal Consistency](https://mlsanity.com/paper/1808.00449/) (2018)
- [Variational Inference of Disentangled Latent Concepts from Unlabeled Observations](https://mlsanity.com/paper/1711.00848/) (2017)
- [Weakly-Supervised Disentanglement Without Compromises](https://mlsanity.com/paper/2002.02886/) (2020)
- [Deep Convolutional Inverse Graphics Network](https://mlsanity.com/paper/1503.03167/) (2015)
- [Weakly Supervised Disentanglement with Guarantees](https://mlsanity.com/paper/1910.09772/) (2019)
- [Disentangling Factors of Variation Using Few Labels](https://mlsanity.com/paper/1905.01258/) (2019)
- [LIMP: Learning Latent Shape Representations with Metric Preservation Priors](https://mlsanity.com/paper/2003.12283/) (2020)
- [Composite Shape Modeling via Latent Space Factorization](https://mlsanity.com/paper/1901.02968/) (2019)
- [Style and Content Disentanglement in Generative Adversarial Networks](https://mlsanity.com/paper/1811.05621/) (2018)
- [Disentangling by Subspace Diffusion](https://mlsanity.com/paper/2006.12982/) (2020)
- [Disentangling Factors of Variation by Mixing Them](https://mlsanity.com/paper/1711.07410/) (2017)
- [Product of Orthogonal Spheres Parameterization for Disentangled Representation Learning](https://mlsanity.com/paper/1907.09554/) (2019)
- [DLGAN: Disentangling Label-Specific Fine-Grained Features for Image Manipulation](https://mlsanity.com/paper/1911.09943/) (2019)

## Built on by (most cited first, 20 of 20)

- [Leveraging sparse and shared feature activations for disentangled representation learning](https://mlsanity.com/paper/2304.07939/) (2023)
- [Unifying Causal Representation Learning with the Invariance Principle](https://mlsanity.com/paper/2409.02772/) (2024)
- [From Bricks to Bridges: Product of Invariances to Enhance Latent Space Communication](https://mlsanity.com/paper/2310.01211/) (2023)
- [Neural Latent Geometry Search: Product Manifold Inference via Gromov-Hausdorff-Informed Bayesian Optimization](https://mlsanity.com/paper/2309.04810/) (2023)
- [Towards a Unified Framework of Contrastive Learning for Disentangled Representations](https://mlsanity.com/paper/2311.04774/) (2023)
- [Diffusion Generative Modeling on Lie Group Representations](https://mlsanity.com/paper/2502.02513/) (2025)
- [Intuitive Shape Editing in Latent Space](https://mlsanity.com/paper/2111.12488/) (2021)
- [A Theory of Initialisation's Impact on Specialisation](https://mlsanity.com/paper/2503.02526/) (2025)
- [Connecting Neural Models Latent Geometries with Relative Geodesic Representations](https://mlsanity.com/paper/2506.01599/) (2025)
- [Mechanistic Independence: A Principle for Identifiable Disentangled Representations](https://mlsanity.com/paper/2509.22196/) (2025)
- [Enriching Disentanglement: From Logical Definitions to Quantitative Metrics](https://mlsanity.com/paper/2305.11512/) (2023)
- [DifAttack++: Query-Efficient Black-Box Adversarial Attack via Hierarchical Disentangled Feature Space in Cross-Domain](https://mlsanity.com/paper/2406.03017/) (2024)
- [Latent Functional Maps: a spectral framework for representation alignment](https://mlsanity.com/paper/2406.14183/) (2024)
- [Transferring disentangled representations: bridging the gap between synthetic and real images](https://mlsanity.com/paper/2409.18017/) (2024)
- [In-Context Semi-Supervised Learning](https://mlsanity.com/paper/2512.15934/) (2025)
- [Invariance-adapted decomposition and Lasso-type contrastive learning](https://mlsanity.com/paper/2210.07413/) (2022)
- [Towards Microstructural State Variables in Materials Systems](https://mlsanity.com/paper/2301.04261/) (2023)
- [Parametrizing Product Shape Manifolds by Composite Networks](https://mlsanity.com/paper/2302.14665/) (2023)
- [Disentangled Latent Spaces Facilitate Data-Driven Auxiliary Learning](https://mlsanity.com/paper/2310.09278/) (2023)
- [Disentangled representations of microscopy images](https://mlsanity.com/paper/2506.20649/) (2025)

## Abstract (arXiv)

We propose a novel approach to disentangle the generative factors of variation underlying a given set of observations. Our method builds upon the idea that the (unknown) low-dimensional manifold underlying the data space can be explicitly modeled as a product of submanifolds. This definition of disentanglement gives rise to a novel weakly-supervised algorithm for recovering the unknown explanatory factors behind the data. At training time, our algorithm only requires pairs of non i.i.d. data samples whose elements share at least one, possibly multidimensional, generative factor of variation. We require no knowledge on the nature of these transformations, and do not make any limiting assumption on the properties of each subspace. Our approach is easy to implement, and can be successfully applied to different kinds of data (from images to 3D surfaces) undergoing arbitrary transformations. In addition to standard synthetic benchmarks, we showcase our method in challenging real-world applications, where we compare favorably with the state of the art.