Definition. Finite Gaussian backgrounds and relabelings [fdip-000G]

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To compare two discrete Gaussian models, specify which data a change of labels must carry. A finite Gaussian background consists of a finite incidence presentation \((V,E,s,h)\), positive weights \(w:E\to \mathbb R_{>0}\), a mass \(m>0\), and a nonempty proper retained subset \(R\subsetneq V\). Put \(U=V\setminus R\); both retained and eliminated carriers are nonempty. The specified reference structures are the counting inner products on vertex and edge fields, the weighted edge inner product, and the induced Borel Euclidean volume \(\lambda _V\) on \(\mathbb R^V\). The orthogonal coordinate split identifies it with the product of the retained and eliminated volumes.

Let \(d:\mathbb R^V\to \mathbb R^E\) be incidence, \((df)(e)=f(h(e))-f(s(e))\), and let \(W\) multiply edge fields by \(w\). The derived operators are \[L=d^\dagger Wd,\qquad Q=L+m^2I>0,\] where \(d^\dagger \) is the counting adjoint. In counting bases, \(d\) has array \(D\) and \(L=D^TWD\). The action \(\mathcal A_Q(f)=b_V(f,Qf)/2\), finite positive normalizer \(Z_Q\), Gaussian law \(\mu _Q\), and retained marginal \(\mu _S\) are derived data. For the 4 split components, \(S=A-BC^{-1}E\); the real orthogonal specialization has \(E=B^\dagger \); the block \(B:Y\to X\) is distinct from incidence.

An exact relabeling to a primed background consists of bijections \(p:V\to V'\), \(q:E\to E'\) satisfying \[\begin {aligned} s'q&=ps,&h'q&=ph,\\ w'q&=w,&m'&=m,\qquad pR=R'. \end {aligned}\] It is the all-positive specialization of a signed endpoint arrow. It preserves the attachment of a loop even when its incidence is zero. The induced maps are \[(Pf)(p(v))=f(v),\qquad (Ja)(q(e))=a(e).\] They preserve both counting pairings, the weighted edge pairing, and by isometry of reference volume the reference volume. Allowing coherent endpoint reversals gives the larger signed relation; it does not change what an exact directed relabeling means.

Evaluation at \(q(e)\) gives \((d'Pf)(q(e))=f(h(e))-f(s(e))=(Jdf)(q(e))\). Weight preservation gives \(W'J=JW\). Thus paired transport yields \[L'=PLP^{-1},\qquad Q'=PQP^{-1}.\] Counting isometry gives \(P^{-1}=P^T\), \(J^{-1}=J^T\) in the coordinate bases, recovering \(D'=JDP^T\) and \(W'=JWJ^T\). The normalized-law theorem carries the laws and moving marginals. Identity, inverse and composition are the directed cases of the proved signed-arrow laws, so existence of an exact relabeling is an equivalence relation on backgrounds.

Two broader comparisons retain less data. The positive-scaling Laplacian relation forgets mass, state and partition, and compares nonzero Laplacians by \(L'=cPLP^{-1}\) with \(c>0\) and a vertex permutation. Its cycle trace ratio still separates two examples. An arbitrary invertible real field map \(y=T\phi \) instead changes precision to \(T^{-T}QT^{-1}\) and the counting pairing to \(y^TT^{-T}T^{-1}z\). Its reference Jacobian is \(dy=|\det T|\,d\phi \). Such a map can transport the normalized law by Jacobian compensation, while losing the reference pairing or retained projection. It is not thereby an enriched background arrow.

The covariance-parameter Gaussian density in [friedli2017statistical, Proposition B.55, p. 505] agrees with these conventions after the inverse-precision theorem identifies \(\Sigma =Q^{-1}\). Its change-of-variables argument does not specify the endpoint, mass or retained-subset data. Mathlib's Matrix.reindexLinearEquiv and Matrix.reindexRingEquiv support the coordinate reindexing identities. The extra background requirements are those stated here.