Cheeger Inequalities for the Persistent Laplacian

Abstract

We study Cheeger-type inequalities for persistent Laplacians associated with inclusions of simplicial complexes $\mathcal{K}\hookrightarrow \mathcal{L}$. We introduce a persistent up $p$-Laplacian ${\Delta }_{q,p,\mathrm{up}}^{\mathcal{K},\mathcal{L}}$ for $p\ge 1$. For $p=2$, this recovers the usual persistent up Laplacian, while for $p=1$ it yields a nonzero persistent Cheeger constant ${\phi }_q^{\mathcal{K},\mathcal{L}}$. We prove a Cheeger-type inequality relating ${\phi }_q^{\mathcal{K},\mathcal{L}}$ to the smallest nonzero eigenvalue of ${\Delta }_{q,\mathrm{up}}^{\mathcal{K},\mathcal{L}}$. This gives a persistent extension of recent work by Jost and Zhang (arXiv:2302.01069). We then study two more structured settings. Under a locally complete $q$-skeleton assumption on $\mathcal{K}$, we extend the complete-skeleton isoperimetric inequality of Parzanchevski--Rosenthal--Tessler (arXiv:1207.0638) to the persistent setting. For orientable $(q+1)$-dimensional pseudomanifolds, we prove a Kron-type reduction of the persistent up Laplacian to a vertex- and edge-weighted graph Laplacian, possibly with Dirichlet boundary terms, and obtain two-sided Cheeger inequalities; this is related to the dual-graph perspective in the work of Steenbergen--Klivans--Mukherjee (arXiv:1209.5091). We also describe the nonzero persistent Cheeger constant ${\phi }_q^{\mathcal{K},\mathcal{L}}$ explicitly in terms of the dual graph in the non-branching pseudomanifold case. Finally, we specialize our two constructions to graph inclusions and compare them with the graph-pair theory of Mémoli--Wan--Wang (arXiv:2012.02808). We establish precise relationships between the two persistent Cheeger constants arising from our constructions and the corresponding constants derived via Kron reduction.

56 pages, 14 figures