A Hilbert space embedding of persistence diagrams and barcodes

What the math gives to ML

The paper gives a constructive, 1-Lipschitz embedding of persistence diagrams and barcodes equipped with Wasserstein-type matching distances into functions on \(\mathbb{N}\times\mathbb{R}\). The transferable asset is that a variable-cardinality combinatorial object becomes a fixed-coordinate functional representation while preserving perturbation bounds, and for \(p=2\) the target is a separable Hilbert space. In neural networks this supports stable persistence-diagram features, losses, and kernels without repeatedly solving diagram matchings during optimization. The most practical first use is a discretized persistence-landscape layer or auxiliary consistency loss whose stability can be compared directly against raw persistence coordinates and standard diagram distances.