Closed-Form Circumcenter Alignment for Unpaired Multimodal Distributions on the Hypersphere

Many multimodal problems supply each modality in a separate cohort, with no subject measured in more than one: the modalities are unpaired, and no matched example exists from which a joint representation could be fit. Prevailing remedies manufacture the missing correspondence, through adversarial matching, contrastive pairs built from labels, or entropic optimal-transport plans, yet the object they produce is recomputed at every step and leaves no explicit reference in the shared space. We instead place a fixed geometric target in closed form. For a label shared by all cohorts, each modality summarises a class by a prototype on a common hypersphere, and the per-modality prototypes of a class are pulled to their circumcenter, the point equidistant from them in their affine hull. The circumcenter follows from the prototypes' geometry alone, so no subject is tied to another, and it persists after training as an explicit anchor, which a counterfactual module uses to reconstruct the modalities a subject lacks and to decide from a single input. Two quantities, computable before training, say which modalities the geometry can help. On Alzheimer's-disease diagnosis across four unpaired modalities (ADNI, DementiaBank), the method leads six unpaired-learning baselines, entropic optimal transport among them, at $93.9\%$ AUROC on amyloid PET and $86.7\%$ on speech, and stays first when transferred to two external cohorts without adaptation.