Centro-affine Poincaré inequality: Unconditional convex bodies
arXiv:2607.20223
2026
Regularization
1 ideas extracted · analyzed Aug 30, 2026
What the math gives to ML
The paper proves a sharp Poincare inequality for smooth unconditional convex bodies equipped with a centro-affine metric and cone-volume measure. Its transferable asset is an explicit anisotropic gradient-energy bound on variance after removing constant and coordinate-affine modes. This can become a geometry-aware regularizer for normalized embeddings or spherical attention directions, with the convex-body metric controlling which feature variations are considered expensive.
Ideas from this paper
Unverified
2026
Add a centro-affine Dirichlet penalty to a neural module whose inputs or outputs lie on a sphere, such as normalized embeddings or attention directions. The penalty measures intrinsic variation under an unconditional convex-body metric while projecting out the constant and coordinate-affine modes excluded by the theorem.
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