# Meta Optimal Transport

**Authors:** Brandon Amos, Samuel Cohen, Giulia Luise, Ievgen Redko  
**arXiv:** [2206.05262](https://arxiv.org/abs/2206.05262)  
**Date:** 2022-06-10  
**Summary page:** https://www.summarizepaper.com/en/paper/2206.05262v2/

## Key points

- Meta Optimal Transport (Meta OT) utilizes amortized optimization to predict optimal transport maps efficiently by leveraging prior knowledge and information.
- The approach aims to improve computational efficiency by avoiding re-solving problems from scratch, demonstrating significant improvements in computational time compared to standard OT solvers.
- Meta OT models are instantiated in discrete and continuous settings across various domains such as grayscale images, spherical data, classification labels, and color palettes.
- The study suggests potential extensions of Meta OT for dynamic OT problems between continuous measures using partially input-convex neural networks.
- The authors emphasize the importance of further research in continuous settings and highlight the promising nature of Meta Optimal Transport for efficiently solving multiple OT problems with shared structure.

## Summary

In their study on Meta Optimal Transport, Brandon Amos, Samuel Cohen, Giulia Luise, and Ievgen Redko explore the use of amortized optimization to predict optimal transport (OT) maps from input measures. Also known as Meta OT, this approach utilizes prior knowledge and information to efficiently solve new OT problems between different measures. By doing so, they aim to improve computational efficiency by avoiding suboptimal re-solving of each problem from scratch. The authors instantiate Meta OT models in both discrete and continuous settings across various domains such as grayscale images, spherical data, classification labels, and color palettes. Through these applications, they demonstrate significant improvements in computational time compared to standard OT solvers. In their conclusions, the authors highlight the potential for further research in continuous settings. They reference recent work on learning solutions for continuous OT problems and suggest extensions of Meta OT for dynamic OT problems between continuous measures using partially input-convex neural networks. The authors express their gratitude to Eugene Vinitsky, Mark Tygert, Mathieu Blondel, Maximilian Nickel, and Muhammad Izzatullah for their valuable insights and discussions. They also acknowledge the essential tools that enabled their work including Python libraries like Hydra and JAX along with Optimal Transport Tools. Overall,<kgd>Meta Optimal Transport</kgd> offers a promising approach for efficiently solving multiple OT problems with shared structure by leveraging <kgd>Past Knowledge and Information</kgd>. The potential extensions into <kgd>Continuous Settings</kgd> open up new avenues for research in optimizing transport solutions across various domains. This study also highlights the importance of <kgd>Amortized Optimization</kgd> in achieving <kgd>Computational Efficiency</kgd> in solving OT problems.