The Gromov-Hausdorff Distance Between Consecutive Spheres

What the math gives to ML

The paper gives an explicit theory for changing the dimension of spherical metric representations while controlling worst-case pairwise angular distortion. The most transferable asset is the compositional rule for synchronized spherical joins: the distortion of the combined construction is the maximum distortion of its factors rather than the sum of their errors. This suggests blockwise hyperspherical adapters for embedding compression, retrieval, or variable-width inference. The sharp lower bound also supplies a principled target for evaluating whether a learned dimension-changing map is close to the best possible.