Geometric Moment Alignment for Domain Adaptation via Siegel Embeddings
This work addresses domain adaptation for machine learning applications like image processing, offering a more principled geometric approach, though it is incremental as it builds on existing moment-matching methods.
The paper tackles distribution shift in unsupervised domain adaptation by aligning first- and second-order moments of source and target distributions using Siegel embeddings and a Riemannian distance on symmetric positive definite matrices, achieving improved performance on image denoising and classification benchmarks.
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.