A faster algorithm of up persistent Laplacian over non-branching simplicial complexes

In this paper we present an algorithm for computing the matrix representation Δq, upK, L of the up persistent Laplacian q, upK, L over a pair of non-branching and orientation-compatible simplicial complexes K L, which has quadratic time complexity. Moreover, we show that the matrix representation Δq, upK, L can be identified as the Laplacian of a weighted oriented hypergraph, which can be regarded as a higher dimensional generalization of the Kron reduction. Finally, we introduce a Cheeger-type inequality with respect to the minimal eigenvalue λₘᵢₙK, L of Δq, upK, L.