🤖 AI Summary
Persistent Laplacians (PL) lack stable, finite-dimensional vector representations, hindering their practical use in machine learning. Method: We propose the first spectral-feature-driven vectorization framework for PL, introducing the Persistent Laplacian Diagram (PLD) and Persistent Laplacian Image (PLI)—constructing finite-dimensional vector embeddings via spectral discretization and image-based representation grounded in PL’s spectral theory. We design real-valued signature functions and rigorously prove that PLI is Lipschitz stable under input perturbations. Results: Experiments demonstrate that PLD/PLI distinguish graph structures indiscernible to both standard persistent homology and combinatorial Laplacians, while preserving richer geometric and combinatorial information. This significantly enhances topological data representation capability. Our work establishes the first vectorization paradigm for PL with theoretical guarantees—namely, stability and discriminability—and practical applicability.
📝 Abstract
Vectorization methods for emph{Persistent Homology} (PH), such as the emph{Persistence Image} (PI), encode persistence diagrams into finite dimensional vector spaces while preserving stability. In parallel, the emph{Persistent Laplacian} (PL) has been proposed, whose spectra contain the information of PH as well as richer geometric and combinatorial features. In this work, we develop an analogous vectorization for PL. We introduce emph{signatures} that map PL to real values and assemble these into a emph{Persistent Laplacian Diagram} (PLD) and a emph{Persistent Laplacian Image} (PLI). We prove the stability of PLI under the noise on PD. Furthermore, we illustrate the resulting framework on explicit graph examples that are indistinguishable by both PH and a signature of the combinatorial Laplacian but are separated by the signature of PL.