Pyramids and Extended Metric Measure Spaces

What the math gives to ML

The paper turns concentration of measure into a concrete representation criterion: a space is concentrated exactly when its equivalence classes of 1-Lipschitz observables are compact, and equivalently when all such observables can be approximated through a common 1-Lipschitz factor map. This suggests a principled bottleneck for neural representations: instead of preserving arbitrary features, preserve the distribution of low-sensitivity observables through a constrained latent factor. A second transferable structure is the 1-Lipschitz partial order between spaces, which can organize multiscale neural representations and token pooling. The strongest practical tests are robustness, compression, and controlled degradation under increasingly coarse representations.