[n] Set {1,…,n} $$\mathbb {R}^{n\times n}$$ The set of real n×n matrices. $$\mathbb {S}^{n}$$ The set of real, symmetric matrices. I The identity matrix of size n×n. 1 The n-dimensional vector whose entries are all equal to 1. σmax(·) Largest singular value of a matrix. $$\mathop {\mathsf {tr}}(\cdot)$$ The trace of a matrix. conv(·) The convex hull of a set. G(V,E) Graph with vertex set V and edge set E. A,B Matrices [ai,j]i,j∈[n],[bi,j]i,j∈[n]. ∥·∥p Operator or entry-wise p-norm. ∥·∥F Frobenius norm. $$\mathbb {P}^{n}$$ Set of permutation matrices of size n×n, c.f. (4) $$\mathbb {W}^{n}$$ Set of doubly stochastic matrices (a.k.a. the Birkhoff polytope) of size n×n, c.f. (5) $$\mathbb {O}^{n}$$ Set of orthofonal matrices (a.k.a. the Stiefel manifold) of size n×n, c.f. (6) Ω, $$\tilde {\Omega }$$ Sets over which a metric is defined. d(x,y) A metric over space Ω. $$\bar {d}(x,y)$$ The symmetric extension of d(x,y). (Ω,d) A metric space. GA,GB Graphs with adjacency matrices A,B. P,W,O n×n matrices. S A closed and bounded subset of $$\ensuremath {\mathbb {R}}^{n\times n}$$. dS(A,B) A class of distance scores defined by minimization (12) over set S. $$d_{\mathbb {P}^{n} }$$ Pseudometric dS, where S is the set of permutation matrices. $$d_{\mathbb {W}^{n} }$$ Pseudometric dS, where S is the set of doubly stochastic matrices. $$d_{\mathbb {O}^{n} }$$ Pseudometric dS, where S is the set of orthogonal matrices. $$\Psi ^{n}_{\tilde {\Omega }}$$ Set of all embeddings from $$[n]\to \tilde {\Omega }$$, where $$(\tilde {\Omega },\tilde {d})$$ is a metric space. ψA,ψB Embeddings in $$\Psi ^{n}_{\tilde {\Omega }}$$ of nodes in graphs GA and GB, respectively. $$D_{\psi _{A},\psi _{B}}$$ n × n matrix of all pairwise distances between images of nodes in GA and GB, under embeddings ψA and ψB.