# Bispectral OT: Dataset Comparison using Symmetry-Aware Optimal Transport

arXiv: https://arxiv.org/abs/2509.20678 · HTML: https://mlsanity.com/paper/2509.20678/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 13 of its 26 references that are in the map.
- First posted on arXiv 25 Sep 2025.

## Builds on (most cited first, 13 of 13)

- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [Joint Distribution Optimal Transportation for Domain Adaptation](https://mlsanity.com/paper/1705.08848/) (2017)
- [Near-linear time approximation algorithms for optimal transport via Sinkhorn iteration](https://mlsanity.com/paper/1705.09634/) (2017)
- [EMNIST: an extension of MNIST to handwritten letters](https://mlsanity.com/paper/1702.05373/) (2017)
- [Geometric Dataset Distances via Optimal Transport](https://mlsanity.com/paper/2002.02923/) (2020)
- [Optimal Transport for Offline Imitation Learning](https://mlsanity.com/paper/2303.13971/) (2023)
- [From optimal transport to generative modeling: the VEGAN cookbook](https://mlsanity.com/paper/1705.07642/) (2017)
- [Structured Optimal Transport](https://mlsanity.com/paper/1712.06199/) (2017)
- [Measuring Generalization with Optimal Transport](https://mlsanity.com/paper/2106.03314/) (2021)
- [Generalizing Point Embeddings using the Wasserstein Space of Elliptical Distributions](https://mlsanity.com/paper/1805.07594/) (2018)
- [Bispectral Neural Networks](https://mlsanity.com/paper/2209.03416/) (2022)
- [Learning Embeddings into Entropic Wasserstein Spaces](https://mlsanity.com/paper/1905.03329/) (2019)
- [The Selective G-Bispectrum and its Inversion: Applications to G-Invariant Networks](https://mlsanity.com/paper/2407.07655/) (2024)

## Abstract (arXiv)

Optimal transport (OT) is a widely used technique in machine learning, graphics, and vision that aligns two distributions or datasets using their relative geometry. In symmetry-rich settings, however, OT alignments based solely on pairwise geometric distances between raw features can ignore the intrinsic coherence structure of the data. We introduce Bispectral Optimal Transport, a symmetry-aware extension of discrete OT that compares elements using their representation using the bispectrum, a group Fourier invariant that preserves all signal structure while removing only the variation due to group actions. Empirically, we demonstrate that the transport plans computed with Bispectral OT achieve greater class preservation accuracy than naive feature OT on benchmark datasets transformed with visual symmetries, improving the quality of meaningful correspondences that capture the underlying semantic label structure in the dataset while removing nuisance variation not affecting class or content.