# Cycle Consistent Probability Divergences Across Different Spaces

arXiv: https://arxiv.org/abs/2111.11328 · HTML: https://mlsanity.com/paper/2111.11328/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 44 references that are in the map.
- First posted on arXiv 22 Nov 2021.

## Builds on (most cited first, 19 of 19)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Enriching Word Vectors with Subword Information](https://mlsanity.com/paper/1607.04606/) (2016)
- [Unpaired Image-to-Image Translation using Cycle-Consistent Adversarial Networks](https://mlsanity.com/paper/1703.10593/) (2017)
- [Word Translation Without Parallel Data](https://mlsanity.com/paper/1710.04087/) (2017)
- [Learning to Discover Cross-Domain Relations with Generative Adversarial Networks](https://mlsanity.com/paper/1703.05192/) (2017)
- [Optimal transport mapping via input convex neural networks](https://mlsanity.com/paper/1908.10962/) (2019)
- [Learning with a Wasserstein Loss](https://mlsanity.com/paper/1506.05439/) (2015)
- [Gromov-Wasserstein Alignment of Word Embedding Spaces](https://mlsanity.com/paper/1809.00013/) (2018)
- [Scalable Gromov-Wasserstein Learning for Graph Partitioning and Matching](https://mlsanity.com/paper/1905.07645/) (2019)
- [Unsupervised Alignment of Embeddings with Wasserstein Procrustes](https://mlsanity.com/paper/1805.11222/) (2018)
- [Sliced Gromov-Wasserstein](https://mlsanity.com/paper/1905.10124/) (2019)
- [The Unbalanced Gromov Wasserstein Distance: Conic Formulation and Relaxation](https://mlsanity.com/paper/2009.04266/) (2020)
- [Regularized Discrete Optimal Transport](https://mlsanity.com/paper/1307.5551/) (2013)
- [Learning Generative Models across Incomparable Spaces](https://mlsanity.com/paper/1905.05461/) (2019)
- [Regularity as Regularization: Smooth and Strongly Convex Brenier Potentials in Optimal Transport](https://mlsanity.com/paper/1905.10812/) (2019)
- [Concentration bounds for linear Monge mapping estimation and optimal transport domain adaptation](https://mlsanity.com/paper/1905.10155/) (2019)
- [Unsupervised Hierarchy Matching with Optimal Transport over Hyperbolic Spaces](https://mlsanity.com/paper/1911.02536/) (2019)
- [Entropic Gromov-Wasserstein between Gaussian Distributions](https://mlsanity.com/paper/2108.10961/) (2021)
- [Unsupervised Correlation Analysis](https://mlsanity.com/paper/1804.00347/) (2018)

## Built on by (most cited first, 5 of 5)

- [Neural Unbalanced Optimal Transport via Cycle-Consistent Semi-Couplings](https://mlsanity.com/paper/2209.15621/) (2022)
- [Unbalanced CO-Optimal Transport](https://mlsanity.com/paper/2205.14923/) (2022)
- [Uncovering Challenges of Solving the Continuous Gromov-Wasserstein Problem](https://mlsanity.com/paper/2303.05978/) (2023)
- [Linear Partial Gromov-Wasserstein Embedding](https://mlsanity.com/paper/2410.16669/) (2024)
- [Sliced Inner Product Gromov-Wasserstein Distances](https://mlsanity.com/paper/2605.08546/) (2026)

## Abstract (arXiv)

Discrepancy measures between probability distributions are at the core of statistical inference and machine learning. In many applications, distributions of interest are supported on different spaces, and yet a meaningful correspondence between data points is desired. Motivated to explicitly encode consistent bidirectional maps into the discrepancy measure, this work proposes a novel unbalanced Monge optimal transport formulation for matching, up to isometries, distributions on different spaces. Our formulation arises as a principled relaxation of the Gromov-Haussdroff distance between metric spaces, and employs two cycle-consistent maps that push forward each distribution onto the other. We study structural properties of the proposed discrepancy and, in particular, show that it captures the popular cycle-consistent generative adversarial network (GAN) framework as a special case, thereby providing the theory to explain it. Motivated by computational efficiency, we then kernelize the discrepancy and restrict the mappings to parametric function classes. The resulting kernelized version is coined the generalized maximum mean discrepancy (GMMD). Convergence rates for empirical estimation of GMMD are studied and experiments to support our theory are provided.