Floating-Body Robust Embedding Core

Attach the method to class prototypes or domain centroids in a representation-learning model. Project each embedding h\in\mathbb R^D to x=Wh\in\mathbb R^d with d=2 or 3, and maintain a reservoir of projected embeddings per class. Approximate each cloud by its convex hull; for an MVP, use a box or an ellipsoid, while the full method uses a polygon or polyhedron. Sample 64--256 directions u, including hull facet normals and random unit vectors. For each u, find t_alpha(u) by bisection so that C_P(u,t_alpha)=alpha vol(P), computing cap volumes by clipping the polygon or polyhedron. Represent the estimated floating body by inequalities u^T c\ge t_alpha(u). Obtain the prototype c by minimizing ||c-mean(P)||^2 subject to these inequalities, or use the displayed hinge penalty with detached thresholds. Use alpha=1/(d+1) as the centerpoint-inspired default; sweep nearby values for robustness. Update the geometry every 100--1000 steps rather than every batch. The paper determines the depth criterion and cap construction; finite reservoirs, direction sampling, and differentiable surrogates are empirical approximations. First test a small ResNet on CIFAR-10 with 2D projection heads, comparing ordinary means, coordinate-trimmed means, and geometric medians. Add 5--20% one-sided outliers or corrupted augmentations. Measure clean and corrupted accuracy, prototype drift, and training stability. Success is improved corrupted accuracy with clean accuracy within 0.2 percentage points of the mean-prototype baseline.