# A Unified Framework for Domain Adaptation using Metric Learning on Manifolds

arXiv: https://arxiv.org/abs/1804.10834 · HTML: https://mlsanity.com/paper/1804.10834/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 28 Apr 2018.

## Built on by (most cited first, 8 of 8)

- [On Riemannian Optimization over Positive Definite Matrices with the Bures-Wasserstein Geometry](https://mlsanity.com/paper/2106.00286/) (2021)
- [Learning with symmetric positive definite matrices via generalized Bures-Wasserstein geometry](https://mlsanity.com/paper/2110.10464/) (2021)
- [Multimodal Attention Merging for Improved Speech Recognition and Audio Event Classification](https://mlsanity.com/paper/2312.14378/) (2023)
- [Using External Off-Policy Speech-To-Text Mappings in Contextual End-To-End Automated Speech Recognition](https://mlsanity.com/paper/2301.02736/) (2023)
- [Task Oriented Dialogue as a Catalyst for Self-Supervised Automatic Speech Recognition](https://mlsanity.com/paper/2401.02417/) (2024)
- [Multi-Stage Multi-Modal Pre-Training for Automatic Speech Recognition](https://mlsanity.com/paper/2403.19822/) (2024)
- [Unsupervised Alignment of Distributional Word Embeddings](https://mlsanity.com/paper/2203.04863/) (2022)
- [Generalized infinite dimensional Alpha-Procrustes based geometries](https://mlsanity.com/paper/2511.09801/) (2025)

## Abstract (arXiv)

We present a novel framework for domain adaptation, whereby both geometric and statistical differences between a labeled source domain and unlabeled target domain can be integrated by exploiting the curved Riemannian geometry of statistical manifolds. Our approach is based on formulating transfer from source to target as a problem of geometric mean metric learning on manifolds. Specifically, we exploit the curved Riemannian manifold geometry of symmetric positive definite (SPD) covariance matrices. We exploit a simple but important observation that as the space of covariance matrices is both a Riemannian space as well as a homogeneous space, the shortest path geodesic between two covariances on the manifold can be computed analytically. Statistics on the SPD matrix manifold, such as the geometric mean of two matrices can be reduced to solving the well-known Riccati equation. We show how the Ricatti-based solution can be constrained to not only reduce the statistical differences between the source and target domains, such as aligning second order covariances and minimizing the maximum mean discrepancy, but also the underlying geometry of the source and target domains using diffusions on the underlying source and target manifolds. A key strength of our proposed approach is that it enables integrating multiple sources of variation between source and target in a unified way, by reducing the combined objective function to a nested set of Ricatti equations where the solution can be represented by a cascaded series of geometric mean computations. In addition to showing the theoretical optimality of our solution, we present detailed experiments using standard transfer learning testbeds from computer vision comparing our proposed algorithms to past work in domain adaptation, showing improved results over a large variety of previous methods.