# A Probabilistic Graph Coupling View of Dimension Reduction

arXiv: https://arxiv.org/abs/2201.13053 · HTML: https://mlsanity.com/paper/2201.13053/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 59 references that are in the map.
- First posted on arXiv 31 Jan 2022.

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

- [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)
- [Understanding How Dimension Reduction Tools Work: An Empirical Approach to Deciphering t-SNE, UMAP, TriMAP, and PaCMAP for Data Visualization](https://mlsanity.com/paper/2012.04456/) (2020)
- [Visualizing Large-scale and High-dimensional Data](https://mlsanity.com/paper/1602.00370/) (2016)
- [Efficient Algorithms for t-distributed Stochastic Neighborhood Embedding](https://mlsanity.com/paper/1712.09005/) (2017)
- [Clustering with t-SNE, provably](https://mlsanity.com/paper/1706.02582/) (2017)
- [An Analysis of the t-SNE Algorithm for Data Visualization](https://mlsanity.com/paper/1803.01768/) (2018)
- [Heavy-tailed kernels reveal a finer cluster structure in t-SNE visualisations](https://mlsanity.com/paper/1902.05804/) (2019)
- [t-SNE-CUDA: GPU-Accelerated t-SNE and its Applications to Modern Data](https://mlsanity.com/paper/1807.11824/) (2018)
- [High-dimensional Gaussian graphical model for network-linked data](https://mlsanity.com/paper/1907.02443/) (2019)

## Built on by (most cited first, 13 of 13)

- [Contrastive Learning Is Spectral Clustering On Similarity Graph](https://mlsanity.com/paper/2303.15103/) (2023)
- [Matrix Information Theory for Self-Supervised Learning](https://mlsanity.com/paper/2305.17326/) (2023)
- [SNEkhorn: Dimension Reduction with Symmetric Entropic Affinities](https://mlsanity.com/paper/2305.13797/) (2023)
- [Distributional Reduction: Unifying Dimensionality Reduction and Clustering with Gromov-Wasserstein](https://mlsanity.com/paper/2402.02239/) (2024)
- [Dimension Reduction with Locally Adjusted Graphs](https://mlsanity.com/paper/2412.15426/) (2024)
- [Assessing and improving reliability of neighbor embedding methods: a map-continuity perspective](https://mlsanity.com/paper/2410.16608/) (2024)
- [Lambda: Learning Matchable Prior For Entity Alignment with Unlabeled Dangling Cases](https://mlsanity.com/paper/2403.10978/) (2024)
- [Towards One Model for Classical Dimensionality Reduction: A Probabilistic Perspective on UMAP and t-SNE](https://mlsanity.com/paper/2405.17412/) (2024)
- [A dimensionality reduction technique based on the Gromov-Wasserstein distance](https://mlsanity.com/paper/2501.13732/) (2025)
- [Kernel-Based Testing for Single-Cell Differential Analysis](https://mlsanity.com/paper/2307.08509/) (2023)
- [Interpolating between Clustering and Dimensionality Reduction with Gromov-Wasserstein](https://mlsanity.com/paper/2310.03398/) (2023)
- [UMAP Is Spectral Clustering on the Fuzzy Nearest-Neighbor Graph](https://mlsanity.com/paper/2602.11662/) (2026)
- [The Rashomon Effect for Visualizing High-Dimensional Data](https://mlsanity.com/paper/2604.00485/) (2026)

## Abstract (arXiv)

Most popular dimension reduction (DR) methods like t-SNE and UMAP are based on minimizing a cost between input and latent pairwise similarities. Though widely used, these approaches lack clear probabilistic foundations to enable a full understanding of their properties and limitations. To that extent, we introduce a unifying statistical framework based on the coupling of hidden graphs using cross entropy. These graphs induce a Markov random field dependency structure among the observations in both input and latent spaces. We show that existing pairwise similarity DR methods can be retrieved from our framework with particular choices of priors for the graphs. Moreover this reveals that these methods suffer from a statistical deficiency that explains poor performances in conserving coarse-grain dependencies. Our model is leveraged and extended to address this issue while new links are drawn with Laplacian eigenmaps and PCA.