# Geometric Moment Alignment for Domain Adaptation via Siegel Embeddings

arXiv: https://arxiv.org/abs/2510.14666 · HTML: https://mlsanity.com/paper/2510.14666/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 45 of its 79 references that are in the map.
- First posted on arXiv 16 Oct 2025.

## Builds on (most cited first, 30 of 45)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Microsoft COCO: Common Objects in Context](https://mlsanity.com/paper/1405.0312/) (2014)
- [Unsupervised Domain Adaptation by Backpropagation](https://mlsanity.com/paper/1409.7495/) (2014)
- [Do ImageNet Classifiers Generalize to ImageNet?](https://mlsanity.com/paper/1902.10811/) (2019)
- [WILDS: A Benchmark of in-the-Wild Distribution Shifts](https://mlsanity.com/paper/2012.07421/) (2020)
- [Learning Transferable Features with Deep Adaptation Networks](https://mlsanity.com/paper/1502.02791/) (2015)
- [Adversarial Discriminative Domain Adaptation](https://mlsanity.com/paper/1702.05464/) (2017)
- [Deep CORAL: Correlation Alignment for Deep Domain Adaptation](https://mlsanity.com/paper/1607.01719/) (2016)
- [Geometric deep learning: going beyond Euclidean data](https://mlsanity.com/paper/1611.08097/) (2016)
- [Poincar'e Embeddings for Learning Hierarchical Representations](https://mlsanity.com/paper/1705.08039/) (2017)
- [Deep Transfer Learning with Joint Adaptation Networks](https://mlsanity.com/paper/1605.06636/) (2016)
- [Deep Domain Confusion: Maximizing for Domain Invariance](https://mlsanity.com/paper/1412.3474/) (2014)
- [New insights and perspectives on the natural gradient method](https://mlsanity.com/paper/1412.1193/) (2014)
- [VisDA: The Visual Domain Adaptation Challenge](https://mlsanity.com/paper/1710.06924/) (2017)
- [Contrastive Adaptation Network for Unsupervised Domain Adaptation](https://mlsanity.com/paper/1901.00976/) (2019)
- [Return of Frustratingly Easy Domain Adaptation](https://mlsanity.com/paper/1511.05547/) (2015)
- [Unsupervised Domain Adaptation with Residual Transfer Networks](https://mlsanity.com/paper/1602.04433/) (2016)
- [Central Moment Discrepancy (CMD) for Domain-Invariant Representation Learning](https://mlsanity.com/paper/1702.08811/) (2017)
- [A Riemannian Network for SPD Matrix Learning](https://mlsanity.com/paper/1608.04233/) (2016)
- [Mind the Class Weight Bias: Weighted Maximum Mean Discrepancy for Unsupervised Domain Adaptation](https://mlsanity.com/paper/1705.00609/) (2017)
- [FixBi: Bridging Domain Spaces for Unsupervised Domain Adaptation](https://mlsanity.com/paper/2011.09230/) (2020)
- [Inductive Moment Matching](https://mlsanity.com/paper/2503.07565/) (2025)
- [YouTube-BoundingBoxes: A Large High-Precision Human-Annotated Data Set for Object Detection in Video](https://mlsanity.com/paper/1702.00824/) (2017)
- [HoMM: Higher-order Moment Matching for Unsupervised Domain Adaptation](https://mlsanity.com/paper/1912.11976/) (2019)
- [A Closer Look at Smoothness in Domain Adversarial Training](https://mlsanity.com/paper/2206.08213/) (2022)
- [Multistep Distillation of Diffusion Models via Moment Matching](https://mlsanity.com/paper/2406.04103/) (2024)
- [Riemannian batch normalization for SPD neural networks](https://mlsanity.com/paper/1909.02414/) (2019)
- [Minimal-Entropy Correlation Alignment for Unsupervised Deep Domain Adaptation](https://mlsanity.com/paper/1711.10288/) (2017)
- [geomstats: a Python Package for Riemannian Geometry in Machine Learning](https://mlsanity.com/paper/1805.08308/) (2018)
- [SPD domain-specific batch normalization to crack interpretable unsupervised domain adaptation in EEG](https://mlsanity.com/paper/2206.01323/) (2022)

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

- [DisRFM: Polar Riemannian Flow Matching for Structure-Preserving Graph Domain Adaptation](https://mlsanity.com/paper/2602.00656/) (2026)
- [Rethinking Time Series Domain Generalization via Structure-Stratified Calibration](https://mlsanity.com/paper/2603.02756/) (2026)

## Abstract (arXiv)

We address the problem of distribution shift in unsupervised domain adaptation with a moment-matching approach. Existing methods typically align low-order statistical moments of the source and target distributions in an embedding space using ad-hoc similarity measures. We propose a principled alternative that instead leverages the intrinsic geometry of these distributions by adopting a Riemannian distance for this alignment. Our key novelty lies in expressing the first- and second-order moments as a single symmetric positive definite (SPD) matrix through Siegel embeddings. This enables simultaneous adaptation of both moments using the natural geometric distance on the shared manifold of SPD matrices, preserving the mean and covariance structure of the source and target distributions and yielding a more faithful metric for cross-domain comparison. We connect the Riemannian manifold distance to the target-domain error bound, and validate the method on image denoising and image classification benchmarks. Our code is publicly available at https://github.com/shayangharib/GeoAdapt.