# Semi-supervised Learning of Pushforwards For Domain Translation & Adaptation

arXiv: https://arxiv.org/abs/2304.08673 · HTML: https://mlsanity.com/paper/2304.08673/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 32 of its 52 references that are in the map.
- First posted on arXiv 18 Apr 2023.

## Builds on (most cited first, 30 of 32)

- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [Unsupervised Domain Adaptation by Backpropagation](https://mlsanity.com/paper/1409.7495/) (2014)
- [Normalizing Flows for Probabilistic Modeling and Inference](https://mlsanity.com/paper/1912.02762/) (2019)
- [Neural Spline Flows](https://mlsanity.com/paper/1906.04032/) (2019)
- [Joint Distribution Optimal Transportation for Domain Adaptation](https://mlsanity.com/paper/1705.08848/) (2017)
- [Why Normalizing Flows Fail to Detect Out-of-Distribution Data](https://mlsanity.com/paper/2006.08545/) (2020)
- [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)
- [Neural Optimal Transport](https://mlsanity.com/paper/2201.12220/) (2022)
- [Unbalanced minibatch Optimal Transport; applications to Domain Adaptation](https://mlsanity.com/paper/2103.03606/) (2021)
- [Large-Scale Optimal Transport and Mapping Estimation](https://mlsanity.com/paper/1711.02283/) (2017)
- [An introduction to domain adaptation and transfer learning](https://mlsanity.com/paper/1812.11806/) (2018)
- [Generative Modeling using the Sliced Wasserstein Distance](https://mlsanity.com/paper/1803.11188/) (2018)
- [Do Neural Optimal Transport Solvers Work? A Continuous Wasserstein-2 Benchmark](https://mlsanity.com/paper/2106.01954/) (2021)
- [Convex Potential Flows: Universal Probability Distributions with Optimal Transport and Convex Optimization](https://mlsanity.com/paper/2012.05942/) (2020)
- [Entropic estimation of optimal transport maps](https://mlsanity.com/paper/2109.12004/) (2021)
- [Robust Optimal Transport with Applications in Generative Modeling and Domain Adaptation](https://mlsanity.com/paper/2010.05862/) (2020)
- [Generative Modeling with Optimal Transport Maps](https://mlsanity.com/paper/2110.02999/) (2021)
- [Plugin Estimation of Smooth Optimal Transport Maps](https://mlsanity.com/paper/2107.12364/) (2021)
- [Composing Normalizing Flows for Inverse Problems](https://mlsanity.com/paper/2002.11743/) (2020)
- [Neural Optimal Transport with General Cost Functionals](https://mlsanity.com/paper/2205.15403/) (2022)
- [AlignFlow: Cycle Consistent Learning from Multiple Domains via Normalizing Flows](https://mlsanity.com/paper/1905.12892/) (2019)
- [An introduction to sampling via measure transport](https://mlsanity.com/paper/1602.05023/) (2016)
- [Near-optimal estimation of smooth transport maps with kernel sums-of-squares](https://mlsanity.com/paper/2112.01907/) (2021)
- [Universal Approximation for Log-concave Distributions using Well-conditioned Normalizing Flows](https://mlsanity.com/paper/2107.02951/) (2021)
- [Guiding the One-to-one Mapping in CycleGAN via Optimal Transport](https://mlsanity.com/paper/1811.06284/) (2018)
- [Optimal Unsupervised Domain Translation](https://mlsanity.com/paper/1906.01292/) (2019)
- [Sliced-Wasserstein normalizing flows: beyond maximum likelihood training](https://mlsanity.com/paper/2207.05468/) (2022)
- [Turning Normalizing Flows into Monge Maps with Geodesic Gaussian Preserving Flows](https://mlsanity.com/paper/2209.10873/) (2022)

## Built on by (most cited first, 2 of 2)

- [Leveraging Optimal Transport via Projections on Subspaces for Machine Learning Applications](https://mlsanity.com/paper/2311.13883/) (2023)
- [Inverse Entropic Optimal Transport Solves Semi-supervised Learning via Data Likelihood Maximization](https://mlsanity.com/paper/2410.02628/) (2024)

## Abstract (arXiv)

Given two probability densities on related data spaces, we seek a map pushing one density to the other while satisfying application-dependent constraints. For maps to have utility in a broad application space (including domain translation, domain adaptation, and generative modeling), the map must be available to apply on out-of-sample data points and should correspond to a probabilistic model over the two spaces. Unfortunately, existing approaches, which are primarily based on optimal transport, do not address these needs. In this paper, we introduce a novel pushforward map learning algorithm that utilizes normalizing flows to parameterize the map. We first re-formulate the classical optimal transport problem to be map-focused and propose a learning algorithm to select from all possible maps under the constraint that the map minimizes a probability distance and application-specific regularizers; thus, our method can be seen as solving a modified optimal transport problem. Once the map is learned, it can be used to map samples from a source domain to a target domain. In addition, because the map is parameterized as a composition of normalizing flows, it models the empirical distributions over the two data spaces and allows both sampling and likelihood evaluation for both data sets. We compare our method (parOT) to related optimal transport approaches in the context of domain adaptation and domain translation on benchmark data sets. Finally, to illustrate the impact of our work on applied problems, we apply parOT to a real scientific application: spectral calibration for high-dimensional measurements from two vastly different environments