arXiv:2608.25145 2026 Geometry 2 ideas extracted · analyzed Aug 29, 2026
The paper provides a principled way to compare a source representation with a one-to-many target representation without penalizing target-side variation that preserves conditional means. Its key transferable asset is replacing pointwise target relations by inner products of barycenters, equivalently searching for an intermediate target measure in convex order below the observed target law. This is useful for prototype-to-instance alignment, weak supervision, and multimodal representation learning where one source item legitimately maps to a distribution of targets. A practical first transfer is a barycentric cross-modal alignment loss, optionally combined with a convex-order refinement constraint that separates semantic mean alignment from within-class diversity.
Replace pointwise cross-modal or prototype-to-instance matching with a loss that compares source pairwise inner products to inner products between target conditional means. A source prototype can align to a cloud of target instances while preserving its semantic barycenter, instead of being forced to match every target instance individually.
Train a conditional generator or set-valued predictor so that stochastic target refinements preserve the barycentric representation required by the source while allowing valid target-side diversity. The regularizer discourages collapse of multiple legitimate outcomes to one point without treating mean-preserving spread as semantic misalignment.