Domain Transfer Becomes Identifiable via a Single Alignment

TL;DR

This paper demonstrates that domain transfer can be uniquely identified with minimal supervision by leveraging a sparsity condition and a single paired sample, improving over previous methods.

Contribution

It introduces a novel identifiability result for domain transfer using just one paired sample under a Jacobian sparsity assumption, with an efficient regularizer for high-dimensional data.

Findings

Theoretical proof of domain transfer identifiability with one paired sample.

Proposed a scalable Jacobian sparsity regularizer for practical learning.

Empirical validation on synthetic and real-world tasks confirms the theory.

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.