# SNEkhorn: Dimension Reduction with Symmetric Entropic Affinities

arXiv: https://arxiv.org/abs/2305.13797 · HTML: https://mlsanity.com/paper/2305.13797/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 59 references that are in the map.
- First posted on arXiv 23 May 2023.

## Builds on (most cited first, 16 of 16)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [UMAP: Uniform Manifold Approximation and Projection for Dimension Reduction](https://mlsanity.com/paper/1802.03426/) (2018)
- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [Barlow Twins: Self-Supervised Learning via Redundancy Reduction](https://mlsanity.com/paper/2103.03230/) (2021)
- [Sinkformers: Transformers with Doubly Stochastic Attention](https://mlsanity.com/paper/2110.11773/) (2021)
- [Efficient Algorithms for t-distributed Stochastic Neighborhood Embedding](https://mlsanity.com/paper/1712.09005/) (2017)
- [Theoretical Foundations of t-SNE for Visualizing High-Dimensional Clustered Data](https://mlsanity.com/paper/2105.07536/) (2021)
- [Clustering with t-SNE, provably](https://mlsanity.com/paper/1706.02582/) (2017)
- [Heavy-tailed kernels reveal a finer cluster structure in t-SNE visualisations](https://mlsanity.com/paper/1902.05804/) (2019)
- [Manifold learning with bi-stochastic kernels](https://mlsanity.com/paper/1711.06711/) (2017)
- [Doubly-Stochastic Normalization of the Gaussian Kernel is Robust to Heteroskedastic Noise](https://mlsanity.com/paper/2006.00402/) (2020)
- [Barnes-Hut-SNE](https://mlsanity.com/paper/1301.3342/) (2013)
- [Doubly Stochastic Subspace Clustering](https://mlsanity.com/paper/2011.14859/) (2020)
- [A Probabilistic Graph Coupling View of Dimension Reduction](https://mlsanity.com/paper/2201.13053/) (2022)
- [Optimal spectral transportation with application to music transcription](https://mlsanity.com/paper/1609.09799/) (2016)
- [Doubly Stochastic Neighbor Embedding on Spheres](https://mlsanity.com/paper/1609.01977/) (2016)

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

- [Distributional Reduction: Unifying Dimensionality Reduction and Clustering with Gromov-Wasserstein](https://mlsanity.com/paper/2402.02239/) (2024)
- [NeuroMax: Enhancing Neural Topic Modeling via Maximizing Mutual Information and Group Topic Regularization](https://mlsanity.com/paper/2409.19749/) (2024)
- [Doubly Stochastic Adaptive Neighbors Clustering via the Marcus Mapping](https://mlsanity.com/paper/2408.02932/) (2024)
- [Inductive Global and Local Manifold Approximation and Projection](https://mlsanity.com/paper/2406.08097/) (2024)
- [Entropic Optimal Transport Eigenmaps for Nonlinear Alignment and Joint Embedding of High-Dimensional Datasets](https://mlsanity.com/paper/2407.01718/) (2024)
- [A dimensionality reduction technique based on the Gromov-Wasserstein distance](https://mlsanity.com/paper/2501.13732/) (2025)
- [SOTAlign: Semi-Supervised Alignment of Unimodal Vision and Language Models via Optimal Transport](https://mlsanity.com/paper/2602.23353/) (2026)
- [Optimal Transport with Adaptive Regularisation](https://mlsanity.com/paper/2310.02925/) (2023)
- [Neuc-MDS: Non-Euclidean Multidimensional Scaling Through Bilinear Forms](https://mlsanity.com/paper/2411.10889/) (2024)
- [Interpolating between Clustering and Dimensionality Reduction with Gromov-Wasserstein](https://mlsanity.com/paper/2310.03398/) (2023)
- [Euclidean Distance Deflation Under High-Dimensional Heteroskedastic Noise](https://mlsanity.com/paper/2507.18520/) (2025)
- [Supervised Distributional Reduction via Optimal Transport and Dependence Maximization](https://mlsanity.com/paper/2605.27619/) (2026)

## Abstract (arXiv)

Many approaches in machine learning rely on a weighted graph to encode the similarities between samples in a dataset. Entropic affinities (EAs), which are notably used in the popular Dimensionality Reduction (DR) algorithm t-SNE, are particular instances of such graphs. To ensure robustness to heterogeneous sampling densities, EAs assign a kernel bandwidth parameter to every sample in such a way that the entropy of each row in the affinity matrix is kept constant at a specific value, whose exponential is known as perplexity. EAs are inherently asymmetric and row-wise stochastic, but they are used in DR approaches after undergoing heuristic symmetrization methods that violate both the row-wise constant entropy and stochasticity properties. In this work, we uncover a novel characterization of EA as an optimal transport problem, allowing a natural symmetrization that can be computed efficiently using dual ascent. The corresponding novel affinity matrix derives advantages from symmetric doubly stochastic normalization in terms of clustering performance, while also effectively controlling the entropy of each row thus making it particularly robust to varying noise levels. Following, we present a new DR algorithm, SNEkhorn, that leverages this new affinity matrix. We show its clear superiority to state-of-the-art approaches with several indicators on both synthetic and real-world datasets.