Block-Cholesky Spaces in CGGMs

• Block-Cholesky spaces (BC-spaces) are defined as linear subspaces of real symmetric matrices partitioned into blocks, enabling tractable Bayesian normalization in colored Gaussian graphical models.
• They leverage combinatorial conditions such as color-perfect elimination ordering and 2-path regularity to guarantee explicit closed-form Diaconis-Ylvisaker integrals for model selection.
• Their structure generalizes decomposable graphs and Bose–Mesner algebras, providing efficient computation and expanded applicability in symmetry-restricted statistical models.

Block-Cholesky spaces (BC-spaces) are a class of linear subspaces of real symmetric matrices structured by block decompositions, introduced to characterize families of colored Gaussian graphical models (CGGMs) that admit explicit closed-form normalizing constants for Bayesian structure learning. These spaces generalize the settings of decomposable graphs and one-color RCOP (random covariance under permutation) models, extending tractability to broader symmetry and sparsity constraints induced by colored graphs. The notion of BC-spaces is closely tied to combinatorial and algebraic conditions on colored graphs, and their analysis yields efficient computation of Diaconis-Ylvisaker integrals central to Gaussian graphical model selection (Chojecki et al., 23 Jan 2026).

1. Formal Structure and Definition

Let $p$ be a positive integer, and partition $\{1,\dots,p\}$ into $r$ ordered blocks of sizes $n_1,\dots,n_r$. Any $x \in \mathrm{Sym}(p)$ is written in $r \times r$ block-matrix form with $X_{kh} \in \mathbb{R}^{n_k \times n_h}$ and $X_{hk} = X_{kh}^\top$. A linear subspace $\mathcal{Z} \subseteq \mathrm{Sym}(p)$ is a BC-space of rank $r$ if:

• (Z0) The identity $I_p \in \mathcal{Z}$, and $\mathcal{Z}$ decomposes as

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

where $M_i(\mathcal{Z})$ consists of block-diagonal matrices supported only on $X_{ii}$ and $L_i(\mathcal{Z})$ consists of matrices supported on the off-diagonal "layer" blocks $(k=i<h)$ and their transposes.

• (Z1) For every $x \in \mathcal{Z}$, the block-lower-triangular truncation

$$\mathrm{BlockTri}(x) = \begin{bmatrix} X_{11} & 0 & \cdots & 0 \ X_{21} & X_{22} & \cdots & 0 \ \vdots & \ddots & \ddots & \vdots \ X_{r1} & \cdots & X_{r,r-1} & X_{rr} \end{bmatrix}$$

satisfies $$\mathrm{BlockTri}(x) \, \mathrm{BlockTri}(x)^\top \in \mathcal{Z}$$.

A BC-space is said to be a diagonally-commutative BC-space (DCBC, also denoted cBC-space) if for all $x, y \in \mathcal{Z}$,

$$\mathrm{BlockDiag}(x)\mathrm{BlockDiag}(y) \in \mathcal{Z}$$

where $\mathrm{BlockDiag}(x)$ extracts the block-diagonal matrix. This implies that the subalgebras $M_i(\mathcal{Z})$ commute under the standard matrix product.

2. Algebraic and Combinatorial Characterization

BC-spaces arise as symmetry-restricted subspaces in CGGMs where precision matrices are both sparse and color-symmetric. Explicitly, for a colored graph $(G, \mathcal{C})$ on $p$ vertices, the RCON-space

$$\mathcal{Z}_G^{\mathcal{C}} = \left\{ K \in \mathrm{Sym}(p) : \begin{array}{l} K_{ij} = 0 \text{ if } \{i,j\} \notin E \ K_{ij} = K_{i'j'} \text{ whenever } \{i,j\}, \{i',j'\} \text{ share the same color} \end{array} \right\}$$

is a BC-space if $(G, \mathcal{C})$ admits a color-perfect elimination ordering (cpeo) and satisfies the 2-path regularity condition.

Each $M_i(\mathcal{Z})$ is a Euclidean Jordan algebra under the product $x \circ y = \frac{1}{2}(xy + yx)$, with identity $e^{(i)} = \mathrm{diag}(0,\dots,I_{n_i},\dots,0)$. The direct-sum and algebraic structure are enforced by the combinatorial properties of the colored graph, in particular by the block associated to the cpeo and regularity of edge colors.

DCBC-spaces further require symmetric 2-path count regularity, ensuring all $M_i$ commute and the algebraic structure is commutative, connecting them to commutative association schemes in the one-color case.

3. Combinatorial Conditions: Color-Perfect Elimination Ordering and 2-Path Regularity

Given a partition of the vertex set into $r$ color classes $V_1,\dots,V_r$, a color-perfect elimination ordering (cpeo) is a permutation $\eta$ of $[r]$ such that for each $i$ and every $v \in V_{\eta_i}$, $v$ remains simplicial (its neighbors form a clique) in the induced subgraph on $V_{\eta_i} \cup \cdots \cup V_{\eta_r}$.

2-path regularity is defined via counts of "color paths": for an extended edge $\{v, w\}$ (possibly a loop) of color $k$,

$$m_{v \to w}(a, b) = |\{ u \in V : c(v,u) = a, c(u,w) = b,\, \mathrm{color}(u) \leq \min(\mathrm{color}(v), \mathrm{color}(w)) \}|$$

with

$$m_{v \leftrightarrow w}(a, b) = m_{v \to w}(a, b) + m_{v \to w}(b,a)$$

It is required that $m_{v \leftrightarrow w}$ is constant across all extended edges of the same color (M1). For the symmetric DCBC case, diagonal color symmetry $m_{v \to w}(a, b) = m_{w \to v}(a, b)$ for $v, w$ sharing a vertex-color (M2) is also imposed.

This framework generalizes decomposable graphs; for example, if a group $\Gamma$ acts generously transitively on each color class and on the edge colors, a peo of $G$ induces a cpeo satisfying all requirements.

4. Closed-Form Evaluation of Diaconis-Ylvisaker Normalizing Constants

For a BC-space $\mathcal{Z}$ of rank $r$ with subalgebras $M_i$ (Jordan rank $d_i$), the generalized Cholesky decomposition writes any $K \in \mathcal{Z} \cap \mathrm{Sym}^+(p)$ as $K = T T^\top$, $$T \in \mathcal{H}^+ \subseteq \mathrm{BlockTri}(\mathcal{Z})$$. Decomposing via Peirce idempotents $c^{(i)}_\alpha$ (with $\alpha=1,\ldots,d_i$, each of matrix-rank $\mu_{i,\alpha}$) and off-diagonal Peirce spaces (multiplicity $m_{i,\alpha}$), the determinant and trace functionals factor:

• $$\det(K) = \prod_{i,\alpha} (t_{i,\alpha}^2/\mu_{i,\alpha})^{\mu_{i,\alpha}}$$
• $\operatorname{tr}(A K)$ reduces to a quadratic form in the Cholesky parameters involving block matrices $\varphi_{i, \alpha}(A)$

The normalization integral

$$I(A,s) = \int_{\mathcal{Z} \cap \mathrm{Sym}^+} (\det K)^s e^{-\operatorname{tr}(A K)} dK$$

splits as a product of independent gamma-Gaussian integrals, admitting the closed-form

$$I(A,s)=e^{-p_\mathcal{Z}s - q_\mathcal{Z}}\prod_{i,\alpha} \Gamma(\mu_{i,\alpha} s + 1 + \tfrac{1}{2} m_{i,\alpha}) \left[\frac{\det \varphi_{i,\alpha}(A)}{\det \psi_{i,\alpha}(A)}\right]^{-(\mu_{i,\alpha} s+1+\tfrac{1}{2} m_{i,\alpha})} (\det \psi_{i,\alpha}(A))^{-1/2}$$

with $p_\mathcal{Z}, q_\mathcal{Z}$ collecting constants and the determinants computable via efficient "pivoted Gram–Cholesky" procedures. For DCBC-spaces, the required eigenstructure is found by diagonalizing a generic linear combination in $M_i$, further reducing computational cost.

5. Example: $2 \times 2$ One-Color Block-Cholesky Space

Consider $p=2$, $r=1$, the space $$\mathcal{Z} = \{ \begin{bmatrix} a & b \ b & a \end{bmatrix} : a, b \in \mathbb{R} \}$$ of dimension 2. Taking $e = \frac{1}{\sqrt{2}}\begin{bmatrix}1&0\0&1\end{bmatrix}$, $f = \frac{1}{\sqrt{2}}\begin{bmatrix}0&1\1&0\end{bmatrix}$, an element is $x = t e + \tau f$. The determinant is $(a-b)(a+b) = (t-\tau)(t+\tau)/2$, so Cholesky variables $t_1 = t-\tau > 0$, $t_2 = t+\tau > 0$ yield the explicit normalizing constant for $I(A,s)$:

$$I(A,s) = \int_{x > 0} \det(x)^s e^{-\operatorname{tr}(A x)} dx = \text{const} \cdot \Gamma(s+1)^2 \cdot a_1^{-(s+1)} a_2^{-(s+1)}$$

where $a_1, a_2$ are the eigenvalues of $A$.

6. Relationship to DCBC-Spaces and Bose–Mesner Algebras

DCBC-spaces are those BC-spaces for which the commutativity condition (Z2) holds. In the case of a single color on the complete graph, $\mathcal{Z}$ coincides with the Bose–Mesner algebra of an association scheme. Here, the idempotents $c_\alpha$ correspond to minimal idempotents of the scheme, and the block decomposition aligns with the adjacency algebra. The explicit normalization formula reduces to the classical Wishart on a commutative algebra, whose structure constants are the association scheme’s intersection numbers and eigenvalues correspond to multiplicities.

Thus, BC-spaces strictly generalize both decomposable graphical model spaces and the commutative Bose–Mesner (one-color) case, recovering the classical results as special instances and extending explicit Bayesian normalization formulas to a broader class of colored graphical models (Chojecki et al., 23 Jan 2026).