Persistent Laplacians: Properties, Algorithms and Implications

We present a thorough study of the theoretical properties and devise efficient algorithms for persistent Laplacian, an extension standard combinatorial Laplacian to setting pairs (or, in more generality, sequences) simplicial complexes $K \hookrightarrow L$, which was recently introduced by Wang, Nguyen, Wei. In particular, analogy with nonpersistent case, we first prove that nullity $q$th $\Delta_q^{K,L}$ equals Betti number inclusion $(K L)$. then initial algorithm finding matrix representation $\Delta_q^{K,L}$, itself helps interpret Laplacian. exhibit novel relationship between notion Schur complement has several important implications. graph it both uncovers link effective resistance leads version Cheeger inequality. This also yields additional, very simple (a of) turn fundamentally different computing pair $K\hookrightarrow L$ can be significantly than algorithms. Finally, Laplacians filtrations establish functoriality stability results their eigenvalues. Our work brings methods from spectral theory, circuit homology together topological view on complexes.