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# Table 1 Notation Summary

[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}. |