Persistence diagrams summarize the topology of data — birth-death pairs marking when topological features (connected components, loops, voids) appear and disappear across scales. But diagrams are awkward inputs for machine learning: they’re multisets of variable size, not vectors, and the natural metric (Wasserstein distance) is expensive to compute.

The paper maps persistence diagrams to functions on the 2-sphere via support functions of lifted zonoids. Each birth-death pair becomes a zone on the sphere; the full diagram becomes a continuous function. The map is linear: superposition of diagrams maps to superposition of functions. And the sphere’s uniform norm controls the partial 1-Wasserstein distance between the original diagrams.

The key advance: bi-continuity. Not just forward stability (small perturbations in diagrams give small perturbations in sphere functions) but inverse continuity — nearby sphere functions come from nearby diagrams. This is the first explicit representation in topological machine learning where the inverse is continuous on the image. You lose nothing essential in the map.

The parameter-free property matters practically. Persistence images require a bandwidth, persistence landscapes require a resolution, kernel methods require a scale — all introduce hyperparameters that affect downstream performance. The sphere representation’s only input is the diagram itself; the norm between base and transformed measures depends solely on the persistence of the input.

Topology as a function on a sphere. The curved surface is a natural home for the planar diagram — the convex geometry of zonoids provides the bridge that the Euclidean embedding doesn’t.