Reproduce “Persistent Laplacians: properties, algorithms and implications” on OpenResearch
Turn your favorite coding agents into research agents that review literature, develop hypotheses, and run experiments.
Universal app for Apple silicon and Intel. Requires macOS 11 or later.
About this paper
Facundo Mémoli, Zhengchao Wan, Yusu Wang
This paper presents a comprehensive study of q-th persistent Laplacians, establishing that their nullity equals q-th persistent Betti numbers and revealing their equivalence to the Schur complement of standard Laplacians. The work provides efficient algorithms for their computation and connects them to concepts like effective resistance and Kron reduction in network theory, alongside proving stability properties for their eigenvalues.
Key insights
- The nullity of the q-th persistent Laplacian precisely equals the q-th persistent Betti number, providing a direct spectral method for measuring persistent topological features.
- The q-th up persistent Laplacian can be expressed as the Schur complement of a principal submatrix of the standard up Laplacian, leading to significantly more efficient computational algorithms.
- For graphs (q=0), the persistent Laplacian is equivalent to the Laplacian of a Kron-reduced graph and preserves effective resistance between vertices or sets, providing intuitive network-theoretic interpretations.
Summary by alphaXiv.
Tip: change arxiv.org to autoarxiv.org in any arXiv abstract link to open the paper here.