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.