Title: Geometric Moment Alignment for Domain Adaptation via Siegel Embeddings
Primary area: learning on graphs and other geometries & topologies
Keywords: domain adaptation, moment matching, Siegel embeddings, SPD geometry, Riemannian manifold

Abstract: 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  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.

TL;DR: we propose a novel and geometric-aware moment matching technique through Siegel embeddings for domain adaptation, and empirically validate the method on standard image benchmark datasets.