A Simplicial Complexes, Cycles, Barcodes

A.1 Background The simplicial complex is a combinatorial data that can be thought of as a higher-dimensional generalization of a graph. Simplicial complex S is a collection of k simplices, which are finite (k + 1) elements subsets in a given set V, for k 0. The collection of simplices S must satisfy the condition that for each σ S, σ S. A simplicial complex consisting only of 0 and 1 simplices is a graph. Let (Γ, m) be a weighted graph with distance-like weights, where m is the symmetric matrix of the weights attached to the edges of the graph Γ. Even though such weighted graphs do not always come from a set of points in metric space, barcodes of weighted graphs have been successfully applied in many situations (networks, fmri, medical data, graph's classification etc). A.2 Simplices, describing discrepancies between the two manifolds Here we gather more details on the construction of sets of simplicies that describe discrepancies between two point clouds P and Q sampled from the two distributions P, Q.