Closing the Modality Gap via Simplex-Constrained Representations
While multimodal contrastive models like CLIP and CLAP successfully learn shared representations, their embeddings typically exhibit a persistent modality gap. We propose addressing this separation through a purely geometric approach: changing the embedding domain from the unit sphere, S^{d-1}, to the probability simplex, Δ^{d-1}. Although both manifolds share the same intrinsic dimension, the simplex acts as a structural prior; its nonnegative, unit-mass constraints serve as a cross-modal calibration that forces both modalities to ground themselves in the same coordinate system. By attaching lightweight softmax adapters and using Total Variation similarity, we reduce the centroid _2 gap by 97--99\% across five multimodal retrieval benchmarks. This confirms that simplicial geometry induces genuine semantic alignment, matching or exceeding the retrieval performance of standard unconstrained models.