Transport based embeddings with topological guarantees

What the math gives to ML

The paper provides a principled replacement for Euclidean point-cloud condensation: represent each sample by a strictly positive probability vector and compare it with other samples using KL divergence. The log-sum-exp potential over KL distances is c-convex, enabling a transport/duality construction whose super-level sets preserve the topology of the original density filtration while collapsing noisy directions. This suggests a neural representation layer in which features are converted to simplex-valued distributions, condensed through KL geometry, and decoded through barycentric or exponential-family coordinates rather than Euclidean projections. The most practical first test is a differentiable KL-transport bottleneck for image or graph embeddings, evaluated both on downstream accuracy and preservation of persistent homology under compression.