[*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. |

*σ*_{max}(·) | 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 [*a*_{i,j}]_{i,j∈[n]},[*b*_{i,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. |

*G*_{A},*G*_{B} | Graphs with adjacency matrices *A*,*B*. |

*P*,*W*,*O* | *n*×*n* matrices. |

*S* | A closed and bounded subset of \(\ensuremath {\mathbb {R}}^{n\times n}\). |

*d*_{S}(*A*,*B*) | A class of distance scores defined by minimization (12) over set *S*. |

\(d_{\mathbb {P}^{n} }\) | Pseudometric *d*_{S}, where *S* is the set of permutation matrices. |

\(d_{\mathbb {W}^{n} }\) | Pseudometric *d*_{S}, where *S* is the set of doubly stochastic matrices. |

\(d_{\mathbb {O}^{n} }\) | Pseudometric *d*_{S}, 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. |

*ψ*_{A},*ψ*_{B} | Embeddings in \(\Psi ^{n}_{\tilde {\Omega }}\) of nodes in graphs *G*_{A} and *G*_{B}, respectively. |

\(D_{\psi _{A},\psi _{B}}\) | *n* × *n* matrix of all pairwise distances between images of nodes in *G*_{A} and *G*_{B}, under embeddings *ψ*_{A} and *ψ*_{B}. |