Metric Geometry of Lebesgue, Wasserstein, and Gromov-Wasserstein Spaces: Submetries, Curvature, and Geodesics

What the math gives to ML

The paper supplies a quotient-space viewpoint for relational data: measurable Z-valued kernels are identified under measure-preserving reparameterizations, and the quotient map onto Z-Gromov-Wasserstein space is a submetry. This means quotient distances can be realized by lifted representatives, allowing invariant objectives and interpolations to be implemented in a larger function space without losing the intrinsic metric. The most transferable constructions are submetry-aware alignment losses for graph and network encoders, and generalized geodesic interpolation obtained by coupling two inputs and interpolating their matched Z-valued entries. These ideas should be tested as alternatives to ad hoc permutation augmentation and naive feature interpolation.