Domain Transfer Becomes Identifiable via a Single Alignment

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.

This paper studies how to reliably translate data from one domain to another, such as converting sketches into images or aligning different types of biological measurements, using mostly unpaired samples from the two domains. This problem is inherently ambiguous: even if the translated samples look valid in the target domain, the model may still fail to preserve the content of individual samples—for example, changing one object or digit into another. Recent approaches reduce this ambiguity by requiring extra attribute information for every sample. In contrast, this paper shows that, under reasonable structural conditions, a single aligned source–target pair can be sufficient to identify the correct translation. The paper also proposes an efficient implementation and validates the idea on image translation and single-cell data.