Stable Vectorization of Persistent Laplacians via Spectral Descriptors

Persistence images vectorize persistence diagrams into stable, finite-dimensional features. Inspired by this idea, we developed a vectorization framework for the spectral information encoded by the persistent Laplacian (PL). Given a scalar signature of a persistent Laplacian, we form a Persistent Laplacian Diagram (PLD) and smooth it into a Persistent Laplacian Image (PLI). We prove a stability theorem for PLIs with respect to the Wasserstein distance between PLDs under an admissibility condition on the signature. Through experiments on MNIST and QM7, we show that PLI with suitable signatures, especially the trace, provides an effective way to extract predictive topological and geometric information from PL, outperforming existing PL-based representations in these settings.