arXiv:2607.16817 2026 Geometry 1 ideas extracted · analyzed Aug 30, 2026
The paper supplies an explicit curvature-dependent distortion coefficient for interpolation on forward Finslerian spaces, together with a Brunn–Minkowski inequality that lower-bounds the measure of geodesic interpolants. The transferable asset is not the specialized sub-Finsler geometry itself, but a principled replacement for uniform Mixup: interpolation weights and augmentation strength can depend on an anisotropic distance and curvature parameter. A practical neural adaptation is to learn a positive Finsler norm in embedding space, use its approximate geodesics to form intermediate examples, and penalize violations of the distortion-weighted volume inequality.
Replace Euclidean Mixup with interpolation in a learned anisotropic embedding metric. Use the paper's distortion coefficient to weight the endpoints and add a consistency term requiring the model's interpolated prediction to respect the geometry-dependent mass allocation.