Umbilic Multinomial Logistic Regression

Multinomial logistic regression (MLR) is the default last-layer classifier in fully supervised recognition, while prototype classifiers are central to few-shot recognition. Although they are usually presented as different principles, we show that both can be organized by a geometric notion of class boundaries. A Euclidean MLR logit is a signed distance to a class hyperplane, so ordinary MLR is already the hyperplane decision-boundary case of Umbilic Multinomial Logistic Regression (UMLR). Classical submanifold geometry supplies the matching spherical decision boundary: in Euclidean space, the complete connected totally umbilic hypersurfaces are hyperplanes and spheres. The spherical UMLR form keeps a prototype-like center but adds a learnable radius; its signed squared-distance form contains squared prototype scoring as the radius-zero case. Thus UMLR is not a separate replacement for MLR or prototypes, but a boundary-aware family that places hyperplane logits, spherical logits, mixed heads, and prototype scoring in one framework. Experiments on low-shot vision head swaps, prototype comparisons, and mixed hyperplane--sphere heads support this organization and show how the different decision boundaries behave across data regimes. UMLR provides a compact geometric language for studying when class evidence is better modeled by flat boundaries, spherical boundaries, or their mixture.