Domain Transfer Becomes Identifiable via a Single Alignment

2026-05-18

AI Summary

This work addresses the non-identifiability of domain translation mappings caused by measure-preserving automorphisms, which leads to content misalignment across domains. Under the assumption that the Jacobian matrix of the mapping exhibits structural sparsity, the authors theoretically prove that a single paired anchor sample together with distribution matching suffices to uniquely identify the true translation mapping—establishing identifiability under weak supervision for the first time and substantially reducing reliance on labeled conditional distributions. To enforce Jacobian sparsity without explicitly computing high-dimensional Jacobians, they propose a stochastic masking-based finite difference regularization technique. Experiments on both synthetic and real-world datasets validate the theoretical claims and demonstrate accurate, scalable cross-domain alignment.

Abstract

Domain transfer (DT) maps source to target distributions and supports tasks such as unsupervised image-to-image translation, single-cell analysis, and cross-platform medical imaging. However, DT is fundamentally ill-posed: push-forward mappings are generally non-identifiable, as measure-preserving automorphisms (MPAs) preserve marginals while altering cross-domain correspondences, leading to content-misaligned translation. Recent work shows that MPAs can be eliminated by jointly transferring multiple corresponding source/target conditional distributions, but supervision signals labeling such conditionals are not always available in practice. We develop an alternative route to DT identifiability. Under a structural sparsity condition on the Jacobian support pattern, we show that distribution matching together with a single paired anchor sample suffices to identify the ground-truth transfer -- requiring substantially less supervision than prior approaches. To enable practical high-dimensional learning, we further propose an efficient Jacobian sparsity regularizer based on randomized masked finite differences, yielding a scalable surrogate without explicit Jacobian evaluation. Empirical results on synthetic and real-world DT tasks validate the theory.

Problem

• domain transfer
• identifiability
• measure-preserving automorphisms
• distribution alignment
• supervision scarcity

Innovation

• domain transfer
• identifiability
• Jacobian sparsity
• anchor sample
• measure-preserving automorphisms