# Distributional Reduction: Unifying Dimensionality Reduction and Clustering with Gromov-Wasserstein

arXiv: https://arxiv.org/abs/2402.02239 · HTML: https://mlsanity.com/paper/2402.02239/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- Builds on 29 of its 99 references that are in the map.
- First posted on arXiv 3 Feb 2024.

## Builds on (most cited first, 29 of 29)

- [Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms](https://mlsanity.com/paper/1708.07747/) (2017)
- [UMAP: Uniform Manifold Approximation and Projection for Dimension Reduction](https://mlsanity.com/paper/1802.03426/) (2018)
- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [Learning Continuous Hierarchies in the Lorentz Model of Hyperbolic Geometry](https://mlsanity.com/paper/1806.03417/) (2018)
- [Riemannian Adaptive Optimization Methods](https://mlsanity.com/paper/1810.00760/) (2018)
- [Optimal Transport for structured data with application on graphs](https://mlsanity.com/paper/1805.09114/) (2018)
- [Gromov-Wasserstein Alignment of Word Embedding Spaces](https://mlsanity.com/paper/1809.00013/) (2018)
- [Geoopt: Riemannian Optimization in PyTorch](https://mlsanity.com/paper/2005.02819/) (2020)
- [Scalable Gromov-Wasserstein Learning for Graph Partitioning and Matching](https://mlsanity.com/paper/1905.07645/) (2019)
- [Convergence Rate of Frank-Wolfe for Non-Convex Objectives](https://mlsanity.com/paper/1607.00345/) (2016)
- [Linear-Time Gromov Wasserstein Distances using Low Rank Couplings and Costs](https://mlsanity.com/paper/2106.01128/) (2021)
- [HoroPCA: Hyperbolic Dimensionality Reduction via Horospherical Projections](https://mlsanity.com/paper/2106.03306/) (2021)
- [CO-Optimal Transport](https://mlsanity.com/paper/2002.03731/) (2020)
- [Generalized Spectral Clustering via Gromov-Wasserstein Learning](https://mlsanity.com/paper/2006.04163/) (2020)
- [Online Graph Dictionary Learning](https://mlsanity.com/paper/2102.06555/) (2021)
- [CO-SNE: Dimensionality Reduction and Visualization for Hyperbolic Data](https://mlsanity.com/paper/2111.15037/) (2021)
- [Minimum-Distortion Embedding](https://mlsanity.com/paper/2103.02559/) (2021)
- [Gromov-Wasserstein Factorization Models for Graph Clustering](https://mlsanity.com/paper/1911.08530/) (2019)
- [Aligning individual brains with Fused Unbalanced Gromov-Wasserstein](https://mlsanity.com/paper/2206.09398/) (2022)
- [Template based Graph Neural Network with Optimal Transport Distances](https://mlsanity.com/paper/2205.15733/) (2022)
- [Hyperbolic Diffusion Embedding and Distance for Hierarchical Representation Learning](https://mlsanity.com/paper/2305.18962/) (2023)
- [A Gromov--Wasserstein Geometric View of Spectrum-Preserving Graph Coarsening](https://mlsanity.com/paper/2306.08854/) (2023)
- [Nested Hyperbolic Spaces for Dimensionality Reduction and Hyperbolic NN Design](https://mlsanity.com/paper/2112.03402/) (2021)
- [A Probabilistic Graph Coupling View of Dimension Reduction](https://mlsanity.com/paper/2201.13053/) (2022)
- [SNEkhorn: Dimension Reduction with Symmetric Entropic Affinities](https://mlsanity.com/paper/2305.13797/) (2023)
- [Doubly Stochastic Neighbor Embedding on Spheres](https://mlsanity.com/paper/1609.01977/) (2016)
- [Block majorization-minimization with diminishing radius for constrained nonsmooth nonconvex optimization](https://mlsanity.com/paper/2012.03503/) (2020)
- [(Probably) Concave Graph Matching](https://mlsanity.com/paper/1807.09722/) (2018)
- [Unified Framework for Spectral Dimensionality Reduction, Maximum Variance Unfolding, and Kernel Learning By Semidefinite Programming: Tutorial and Survey](https://mlsanity.com/paper/2106.15379/) (2021)

## Built on by (most cited first, 12 of 12)

- [On Probabilistic Embeddings in Optimal Dimension Reduction](https://mlsanity.com/paper/2408.02433/) (2024)
- [A dimensionality reduction technique based on the Gromov-Wasserstein distance](https://mlsanity.com/paper/2501.13732/) (2025)
- [Constrained Sliced Wasserstein Embedding](https://mlsanity.com/paper/2506.02203/) (2025)
- [Structure-Preserving Multi-View Embedding Using Gromov-Wasserstein Optimal Transport](https://mlsanity.com/paper/2604.02610/) (2026)
- [Linear Partial Gromov-Wasserstein Embedding](https://mlsanity.com/paper/2410.16669/) (2024)
- [Graph-based Clustering Revisited: A Relaxation of Kernel k-Means Perspective](https://mlsanity.com/paper/2509.18826/) (2025)
- [Gromov-Wasserstein Methods for Multi-View Relational Embedding and Clustering](https://mlsanity.com/paper/2604.23912/) (2026)
- [Group Contrastive Learning for Weakly Paired Multimodal Data](https://mlsanity.com/paper/2602.04021/) (2026)
- [On sparsity, extremal structure, and monotonicity properties of Wasserstein and Gromov-Wasserstein optimal transport plans](https://mlsanity.com/paper/2602.16265/) (2026)
- [Bridging Maximum Likelihood and Optimal Transport for Efficient Inference and Model Selection in Stochastic Block Models](https://mlsanity.com/paper/2605.28488/) (2026)
- [Network Learning with Semi-relaxed Gromov-Wasserstein](https://mlsanity.com/paper/2606.02223/) (2026)
- [Gromov-Wasserstein Distillation for Inductive Multi-View Embedding](https://mlsanity.com/paper/2609.40047/) (2026)

## Abstract (arXiv)

Unsupervised learning aims to capture the underlying structure of potentially large and high-dimensional datasets. Traditionally, this involves using dimensionality reduction (DR) methods to project data onto lower-dimensional spaces or organizing points into meaningful clusters (clustering). In this work, we revisit these approaches under the lens of optimal transport and exhibit relationships with the Gromov-Wasserstein problem. This unveils a new general framework, called distributional reduction, that recovers DR and clustering as special cases and allows addressing them jointly within a single optimization problem. We empirically demonstrate its relevance to the identification of low-dimensional prototypes representing data at different scales, across multiple image and genomic datasets.