UOT-Gap: A Variational Principle for the Modality Gap in Vision-Language Models via Unbalanced Optimal Transport

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Vision-language models such as CLIP embed images and text in a shared space, where modality-specific distributions often remain separated. Existing accounts connect this modality gap to initialization, contrastive dynamics, and information imbalance, while its distributional and pairwise contributions to retrieval remain unresolved. We introduce UOT-Gap, a training-free variational diagnostic that models frozen image and text embeddings with unbalanced entropic optimal transport (UOT). The UOT optimum separates transport, coupling complexity, and marginal mass variation; a complementary pair-aware residual compares observed image-caption pairs with the UOT soft matching. On Flickr8K and COCO-1K with frozen CLIP, OpenCLIP, and SigLIP encoders, caption degradation reduces Flickr8K Recall@1 from 0.559 to 0.003. Across six dataset-model conditions, the pair-aware residual tracks retrieval degradation with mean absolute Spearman 0.973, compared with 0.392 for the mean gap. The association remains stable across five random COCO-1K subsets at $0.954\pm0.026$, with a minimum of 0.943. UOT barycentric updates reduce the transport objective while degrading retrieval, distinguishing geometric objective descent from task improvement. These results establish UOT-Gap as a diagnostic for caption quality, modality alignment, and retrieval robustness.