DCBC-spaces: Diagonally Commutative Block-Cholesky Spaces

• DCBC-spaces are defined as linear subspaces of real symmetric matrices with block structure satisfying both triangular and diagonal closure conditions.
• They provide closed-form solutions for normalizing constants in colored Gaussian graphical models, facilitating efficient Bayesian model selection.
• DCBC-spaces bridge algebraic structures and combinatorial graph properties, extending tractability beyond classical decomposable and RCOP models.

Diagonally Commutative Block-Cholesky spaces (DCBC-spaces) form a class of linear subspaces in the space of real symmetric matrices, identified by jointly algebraic and combinatorial criteria. They arise naturally in the study of colored Gaussian graphical models (CGGMs), particularly in the context of Bayesian model selection with symmetry constraints and sparsity patterns. The primary significance of DCBC-spaces lies in providing closed-form solutions for normalizing constants associated with certain integrals critical for statistical inference, expanding the tractable cases well beyond classical decomposable or RCOP models (Chojecki et al., 23 Jan 2026).

1. Algebraic Definition of BC- and DCBC-spaces

Let $p = \sum_{i=1}^r n_i$ for positive integers $n_i$ and $r$. Consider the space $\Sym(p)$ of $p\times p$ real symmetric matrices, with each element $X$ expressed in $r\times r$ block form: $X_{ij} \in \R^{n_i\times n_j}$ with $X_{ji}=X_{ij}^\top$. Define

• $BlockTri(X)$ as the block lower-triangular part of $X$;
• $BlockDiag(X)$ as the block-diagonal part of $X$.

A linear subspace $\Zsp \subset \Sym(p)$ is a Block-Cholesky space (BC-space) of rank $r$ if:

• (Z0) $I_p \in \Zsp$ and there is a direct sum decomposition:

$\Zsp = \Big(\bigoplus_{i=1}^r M_i\Big) \oplus \Big(\bigoplus_{i=1}^{r-1} L_i\Big)$

with

$M_i = \{X\in\Zsp : X_{jk}=0 \text{ unless } j=k=i\}, \quad L_i = \{X\in\Zsp : X_{jk}=0 \text{ unless }\{j,k\}=\{i,\ell\} \text{ for some } \ell > i\}.$

• (Z1) Closure under block-Cholesky multiplication: $BlockTri(X)BlockTri(X)^\top \in \Zsp$ for all $X \in \Zsp$.

$\Zsp$ is a Diagonally Commutative Block-Cholesky space (DCBC-space) if, in addition,

$BlockDiag(X)BlockDiag(Y) \in \Zsp, \quad \forall X,Y \in \Zsp.$

This last property enforces block-diagonal multiplicative closure and commutativity among the $M_i$ blocks, rendering each $M_i$ a commutative Jordan algebra.

2. Algebraic Criteria and Structural Characterization

The characterization of BC- and DCBC-spaces is given by closure properties:

• A linear subspace $\Zsp$ with the decomposition above is a BC-space if and only if $BlockTri(J^k)BlockTri(J^\ell)^\top \in \Zsp$ for all basis elements $J^k, J^\ell$.
• For DCBC-spaces, closure under block-diagonal products: $J^k J^\ell \in \Zsp$ for all $J^k, J^\ell \in BlockDiag(\Zsp)$, and $BlockDiag(X)BlockDiag(Y) = BlockDiag(Y)BlockDiag(X) \in \Zsp$.

A direct implication is that, for DCBC-spaces, each block $M_i$ is diagonalizable and forms a commutative Jordan algebra, akin to Bose–Mesner algebras.

3. Combinatorial Construction: CER Graphs and Symmetric CER Graphs

Every colored undirected graph $(G, \mathcal{C})$ on $p$ vertices defines an RCON-space:

$\Zsp_G^\mathcal{C} = \{K \in \Sym(p): K_{ij}=0 \text{ if } \{i,j\}\notin E(G), \; K_{ij}=K_{i'j'} \text{ if } \{i,j\},\{i',j'\} \text{ share a color}\}$

BC- and DCBC-spaces correspond to combinatorial objects known as Color–Elimination–Regular (CER) graphs and symmetric CER graphs, respectively.

The construction requirements are:

• Color-Perfect Elimination Ordering (cPEO): A permutation of the vertex-color classes such that, in the induced linear vertex ordering, each vertex of a given color is simplicial in the induced subgraph on its color class and those after it.
• 2-Path Regularity: For the given cPEO, the multisets counting colored 2-paths (using specific edge and vertex colorings) between any two vertices with the same edge color must coincide across the graph (with symmetry conditions for the symmetric CER case).

The main theorem establishes:

• For any CER graph $(G, \mathcal{C})$, the associated RCON-space $\Zsp_G^\mathcal{C}$ is a BC-space.
• If $(G, \mathcal{C})$ is symmetric CER, $\Zsp_G^\mathcal{C}$ is a DCBC-space.

This construction strictly includes all decomposable graphs (in the classical, uncolored case) and encompasses all RCOP models over decomposable skeletons.

4. Closed-Form Expressions for Normalizing Constants

For $\Zsp$ a BC-space of rank $r$, explicit closed-form formulas are derived for the integral

$I(s;A) = \int_{\Zsp \cap \Sym^+(p)} (\det x)^s \, e^{-\tr(Ax)} \, dx$

where $A \in \Sym^+(p)$. The convergence criterion is

$$s > \max_{i,\alpha}\Bigl\{-\frac{2+m_{i,\alpha}}{2\,\mu_{i,\alpha}}\Bigr\}$$

with structure constants $d_i=\dim M_i$, $\mu_{i,\alpha}=\mathrm{rank}(c_\alpha^{(i)})$, $m_{i,\alpha}=\dim \mathfrak{h}_{i,\alpha}$. The formula for $I(s;A)$ factorizes as a product over $i,\alpha$ of Gamma-functions and determinants of derived matrices $\phi_{i,\alpha}(A), \psi_{i,\alpha}(A)$, with pre-factors $p_{\Zsp}$ and $q_{\Zsp}$ computable from the structure constants. In DCBC-spaces, computation of determinants and block inverses reduces to operations of cubic complexity in the block sizes, making the formula highly tractable (Chojecki et al., 23 Jan 2026).

5. Explicit Example and Computational Illustration

Consider $p=2$ and $\Zsp = \big\{\text{diag}(a,b): a,b\in\R\big\}$ (the simplest nontrivial DCBC-structure, $r=2$, $n_1=n_2=1$). Then $\Zsp\cap \Sym^+(2) = \{\text{diag}(a,b):a>0, b>0\}$, and

$$I(s;A) = \int_{a,b>0} a^s b^s e^{-\alpha a - \beta b} da\,db = \Gamma(s+1) \alpha^{-s-1} \Gamma(s+1) \beta^{-s-1}$$

for $A=\text{diag}(\alpha,\beta)$. This reflects the block-wise Gamma-factorization and exemplifies the general formula in the DCBC context.

6. Relationship to Bose–Mesner Algebras and Association Schemes

Any DCBC-space inherently has all $M_i$ as commutative Jordan algebras. In the extremal scenario (single vertex-color class with a complete graph), symmetric CER graphs coincide with symmetric association schemes, and their corresponding matrix algebras with Bose–Mesner algebras. The 2-path regularity imposed combinatorially directly matches intersection numbers in association schemes. In this regime, the normalizing constant reduces to well-known product formulas indexed by idempotents of the Bose–Mesner algebra, linking the theory of DCBC-spaces to the rich structure of association schemes (Chojecki et al., 23 Jan 2026).

7. Context and Extension Beyond Classical Models

The DCBC-space framework provides a unified generalization of Gaussian graphical models with decomposable skeletons and RCOP symmetry models, expanding to larger families of colored graphical models. It enables closed-form and efficiently computable normalization for significantly broader classes of precision matrix spaces, facilitating Bayesian inference and structure learning in high-dimensional colored graphical models. In particular, the combinatorial (CER) and algebraic (commutative block) requirements admit tractable integral evaluation by leveraging block-Cholesky structural decompositions and Gamma-type product formulas (Chojecki et al., 23 Jan 2026).