# Functional optimal transport: map estimation and domain adaptation for functional data

arXiv: https://arxiv.org/abs/2102.03895 · HTML: https://mlsanity.com/paper/2102.03895/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 39 of its 108 references that are in the map.
- First posted on arXiv 7 Feb 2021.

## Builds on (most cited first, 30 of 39)

- [Model-Agnostic Meta-Learning for Fast Adaptation of Deep Networks](https://mlsanity.com/paper/1703.03400/) (2017)
- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [Unsupervised Domain Adaptation by Backpropagation](https://mlsanity.com/paper/1409.7495/) (2014)
- [Domain Randomization for Transferring Deep Neural Networks from Simulation to the Real World](https://mlsanity.com/paper/1703.06907/) (2017)
- [Wasserstein GAN](https://mlsanity.com/paper/1701.07875/) (2017)
- [Optimal Transport for Domain Adaptation](https://mlsanity.com/paper/1507.00504/) (2015)
- [Input Convex Neural Networks](https://mlsanity.com/paper/1609.07152/) (2016)
- [Attentive Neural Processes](https://mlsanity.com/paper/1901.05761/) (2019)
- [DeepJDOT: Deep Joint Distribution Optimal Transport for Unsupervised Domain Adaptation](https://mlsanity.com/paper/1803.10081/) (2018)
- [Optimal transport mapping via input convex neural networks](https://mlsanity.com/paper/1908.10962/) (2019)
- [RoboTurk: A Crowdsourcing Platform for Robotic Skill Learning through Imitation](https://mlsanity.com/paper/1811.02790/) (2018)
- [Stochastic Optimization for Large-scale Optimal Transport](https://mlsanity.com/paper/1605.08527/) (2016)
- [Geometric Dataset Distances via Optimal Transport](https://mlsanity.com/paper/2002.02923/) (2020)
- [Using Simulation and Domain Adaptation to Improve Efficiency of Deep Robotic Grasping](https://mlsanity.com/paper/1709.07857/) (2017)
- [Large-Scale Optimal Transport and Mapping Estimation](https://mlsanity.com/paper/1711.02283/) (2017)
- [Improving GANs Using Optimal Transport](https://mlsanity.com/paper/1803.05573/) (2018)
- [Gaussian Processes for Data-Efficient Learning in Robotics and Control](https://mlsanity.com/paper/1502.02860/) (2015)
- [Unsupervised Alignment of Embeddings with Wasserstein Procrustes](https://mlsanity.com/paper/1805.11222/) (2018)
- [Convex Potential Flows: Universal Probability Distributions with Optimal Transport and Convex Optimization](https://mlsanity.com/paper/2012.05942/) (2020)
- [Smooth and Sparse Optimal Transport](https://mlsanity.com/paper/1710.06276/) (2017)
- [Multiple Interactions Made Easy (MIME): Large Scale Demonstrations Data for Imitation](https://mlsanity.com/paper/1810.07121/) (2018)
- [Scalable Computations of Wasserstein Barycenter via Input Convex Neural Networks](https://mlsanity.com/paper/2007.04462/) (2020)
- [Multilevel Clustering via Wasserstein Means](https://mlsanity.com/paper/1706.03883/) (2017)
- [Outlier-Robust Optimal Transport](https://mlsanity.com/paper/2012.07363/) (2020)
- [Assistive Gym: A Physics Simulation Framework for Assistive Robotics](https://mlsanity.com/paper/1910.04700/) (2019)
- [Convergence and Concentration of Empirical Measures under Wasserstein Distance in Unbounded Functional Spaces](https://mlsanity.com/paper/1804.10556/) (2018)
- [Hierarchical Optimal Transport for Multimodal Distribution Alignment](https://mlsanity.com/paper/1906.11768/) (2019)
- [Towards Optimal Transport with Global Invariances](https://mlsanity.com/paper/1806.09277/) (2018)
- [Improving Sequence-to-Sequence Learning via Optimal Transport](https://mlsanity.com/paper/1901.06283/) (2019)
- [On Scalable and Efficient Computation of Large Scale Optimal Transport](https://mlsanity.com/paper/1905.00158/) (2019)

## Built on by (most cited first, 7 of 7)

- [Neural Monge Map estimation and its applications](https://mlsanity.com/paper/2106.03812/) (2021)
- [Curriculum Reinforcement Learning using Optimal Transport via Gradual Domain Adaptation](https://mlsanity.com/paper/2210.10195/) (2022)
- [Scalable Optimal Transport Methods in Machine Learning: A Contemporary Survey](https://mlsanity.com/paper/2305.05080/) (2023)
- [GeoECG: Data Augmentation via Wasserstein Geodesic Perturbation for Robust Electrocardiogram Prediction](https://mlsanity.com/paper/2208.01220/) (2022)
- [PhysioMTL: Personalizing Physiological Patterns using Optimal Transport Multi-Task Regression](https://mlsanity.com/paper/2203.12595/) (2022)
- [Interpolation for Robust Learning: Data Augmentation on Wasserstein Geodesics](https://mlsanity.com/paper/2302.02092/) (2023)
- [On Hypothesis Transfer Learning of Functional Linear Models](https://mlsanity.com/paper/2206.04277/) (2022)

## Abstract (arXiv)

We introduce a formulation of optimal transport problem for distributions on function spaces, where the stochastic map between functional domains can be partially represented in terms of an (infinite-dimensional) Hilbert-Schmidt operator mapping a Hilbert space of functions to another. For numerous machine learning tasks, data can be naturally viewed as samples drawn from spaces of functions, such as curves and surfaces, in high dimensions. Optimal transport for functional data analysis provides a useful framework of treatment for such domains. { Since probability measures in infinite dimensional spaces generally lack absolute continuity (that is, with respect to non-degenerate Gaussian measures), the Monge map in the standard optimal transport theory for finite dimensional spaces may not exist. Our approach to the optimal transport problem in infinite dimensions is by a suitable regularization technique -- we restrict the class of transport maps to be a Hilbert-Schmidt space of operators.} To this end, we develop an efficient algorithm for finding the stochastic transport map between functional domains and provide theoretical guarantees on the existence, uniqueness, and consistency of our estimate for the Hilbert-Schmidt operator. We validate our method on synthetic datasets and examine the functional properties of the transport map. Experiments on real-world datasets of robot arm trajectories further demonstrate the effectiveness of our method on applications in domain adaptation.