Sparse Cholesky Decomposition

• Sparse Cholesky decomposition is a factorization method that computes lower-triangular matrices with prescribed zeros to reveal conditional independence in positive definite settings.
• It leverages penalized likelihood, regression frameworks, and randomized algorithms to ensure scalability, numerical stability, and positive definiteness.
• Applications include high-dimensional statistics, spatial modeling, PDE solvers, and Gaussian process regression for efficient, interpretable inference.

Sparse Cholesky decomposition refers to any method or framework by which Cholesky factors (lower-triangular or block-lower-triangular matrices) of covariance, precision, or related positive definite matrices can be computed and/or estimated with a prescribed pattern of structural zeros, thereby enabling computational scalability, model interpretability, and direct encoding of conditional independence or graphical constraints. Numerous modern applications in high-dimensional statistics, spatial statistics, numerical PDEs, and machine learning leverage the algebraic, graph-theoretic, and optimization properties of sparse Cholesky representations. Technical advances encompass penalized likelihood estimation, hierarchical factorization, KL-optimal approximation, randomized algorithms, and block-structured models.

1. Mathematical Foundations of Sparse Cholesky Decomposition

Sparse Cholesky methods impose structured zero-patterns on the Cholesky factors of a symmetric positive definite matrix $\Sigma$ or its inverse. These patterns typically correspond to conditional independence assumptions or underlying graphical models:

• Standard Cholesky/Penalty regression interpretation: For a covariance $\Sigma\in\mathbb R^{p\times p}$, the Cholesky factor $T$ (lower-triangular with $T_{ii}>0$) parameterizes $\Sigma=TT^\top$. Sparsity in $T_{ij}$ for $j<i$ encodes that variable $i$ is conditionally independent of variable $j$ given variables $1, \ldots, j-1$ (Córdoba et al., 2020).
• Inverse Cholesky (precision matrix estimation): The inverse covariance $\Omega = \Sigma^{-1}$ admits a block-lower-triangular factorization:

$\Omega=T^\top D^{-1} T$

where $T$ is unit block-lower-triangular and $D$ is block-diagonal positive definite (Kang et al., 2023). In this setting, sparsity in $T$ or in the blocks of $D^{-1}$ encodes directed conditional dependencies.

• Graph-theoretic characterization: For specified chordal or homogeneous graphs $G$ and vertex ordering $\sigma$, sparse Cholesky decomposition preserves zeros in both $\Sigma$ and the Cholesky factor $L$ (and possibly $L^{-1}$) exactly when $G$ satisfies certain ancestral or elimination properties (Khare et al., 2011).

Table: Cholesky Formulations in Sparse Models

Factorization	Sparsity Pattern	Model/Graph Type
$\Sigma = TT^\top$	$T_{ij}=0$ for $j<i$	Ordered conditional independence
$\Sigma^{-1} = T^\top D^{-1}T$	block zeros in $T$ and $D^{-1}$	Partially ordered blocks
$A = LDL^\top$	$L_{ij}=0$ (specified)	Chordal/homogeneous graphs

2. Sparse Cholesky Estimation in High Dimensions

Penalized likelihood and regression-based frameworks have become standard for high-dimensional sparse estimation:

• Block Cholesky Decomposition (BCD): Variables are partitioned into $M$ ordered groups. For each block $j=2,\ldots,M$,

$$X^{(j)} = \sum_{i<j}A_{ji}\,X^{(i)} + \varepsilon_j$$

with $A_{ji}$ (regression blocks) and $D_j=\mathrm{Cov}(\varepsilon_j)\succ 0$ estimated via $\ell_1$ penalties (Kang et al., 2023). The penalized log-likelihood is:

$$L_\lambda(A, D^{-1}) = L(A, D^{-1}) + \lambda_1 \sum_{j=2}^M\|A_j\|_1 + \lambda_2\sum_{j=1}^M\|(D_j^{-1})\|_1^{-}$$

This delivers guaranteed positive-definite precision estimates and exact block-sparsity.

• ADMM ensemble/ordering-invariant approaches: For cases where variable ordering is ambiguous, permutation ensembles with sparse thresholding (hard or soft) are used to form ordering-invariant precision or covariance estimates (Kang et al., 2018, Kang et al., 2017). The center estimator minimizes average Frobenius distance to MCD estimates over permutations, penalized for off-diagonal sparsity.

Table: Sparse Estimation Algorithms

Framework	Optimization Problem	Guarantees
BCD (Kang et al., 2023)	Penalized block regression + block graphical lasso	$\widehat\Omega \succ 0$
MCD ensemble (Kang et al., 2018)	Frobenius penalized ensemble ADMM over random orderings	Convergence, consistency
Convex sparse Cholesky (Khare et al., 2016)	Jointly convex penalized likelihood in $L$	PD, global minimizer

3. Algorithmic Strategies and Computational Complexity

Sparse Cholesky algorithms are tailored to scalable computation in large systems:

• Block coordinate descent: BCD updates each block’s regression coefficients and residual block-covariances parallely; standard lasso and graphical lasso subroutines are used, with convergence to a local minimum (Kang et al., 2023).
• Hierarchical Sparse Cholesky for Spatio-Temporal Models: Vecchia-type factorizations enforce sparsity via conditioning sets and hierarchical orderings; incomplete/sparse Cholesky factorization is $O(n m^2)$ time for grid size $n$ and Markov size $m$ (Jurek et al., 2022, Jurek et al., 2020).
• KL-divergence minimization with optimal sparsity pattern: For kernel matrices/Green’s functions, sparse inverse Cholesky factors are constructed column-wise in closed form for any prescribed sparsity pattern via KL-optimality (Schäfer et al., 2020). With supernode aggregation, complexity is $O(N \log^{2d}(N/\epsilon))$ for accuracy $\epsilon$ in $d$ dimensions.
• Randomized Cholesky (RCHOL) and sparsified solvers: For SDD/Laplacian matrices, randomized clique-sampling yields Cholesky factors with $O(n)$ fill and $O(n \log n)$ time (Chen et al., 2020, Lee et al., 2015). Parallel factorization based on nested dissection achieves multi-core scalability.

4. Theoretical Guarantees: Positive Definiteness, Consistency, and Error Bounds

Sparse Cholesky methodology enjoys rigorous statistical and numerical guarantees:

• Positive definiteness: Block Cholesky algorithms and penalized-determinant formulations guarantee the estimated inverse covariance is strictly positive definite for any sample size, due to the block-lower-triangular and diagonal constraint structure (Kang et al., 2023, Khare et al., 2016).
• High-dimensional convergence rates: Under regularity (bounded eigenvalues, sparsity size, proper scaling of penalties), estimation errors in Frobenius norm decay as:

$$\|\widehat\Omega-\Omega_0\|_F = O_p\left(\sqrt{\frac{s_T\log p + \sum_j (s_{D_j}+p_j)\log p_j}{n}}\right)$$

for block sparsity $s_T$ and sample size $n$ (Kang et al., 2023).

• KL-divergence and minimax optimality: KL-optimal sparse Cholesky approximations minimize the divergence to the true covariance matrix given any fixed sparsity pattern; for elliptic kernel matrices decay rates in the off-diagonal of the true inverse ensure exponential accuracy with logarithmic interaction radius (Schäfer et al., 2020).
• Model selection consistency: ADMM ensemble and ChoSelect methods recover the true Cholesky support (graph structure) under polynomial sample regimes and restricted eigenvalue conditions (Kang et al., 2018, Verzelen, 2010).

5. Graph-Based Insights and Structural Implications

Graph-theoretic properties control the fidelity of sparse Cholesky decompositions:

• Chordal graphs and elimination orderings: Zeros in a positive definite matrix $A$ and its Cholesky factor $L$ are preserved equivariantly only for chordal graphs under a perfect elimination ordering (Khare et al., 2011).
• Homogeneous graphs and simultaneous preservation: For homogeneous (co-chordal) graphs with a Hasse-tree elimination scheme, both $L$ and $L^{-1}$ retain the exact prescribed zero structure, admitting clique-determinant characterizations of the inverse covariance (Khare et al., 2011).
• Ordering dependence and modern solutions: Standard MCD and Cholesky regression estimators, banding, or lasso, depend critically on variable ordering. Ensemble, Frobenius-center, and block-decomposition methods mitigate this dependence via randomization or use of partial information (Kang et al., 2018, Kang et al., 2023).

6. Applications and Methodological Extensions

Sparse Cholesky factorization techniques are fundamental in multiple domains:

• Spatial, spatio-temporal, and state-space modeling: Scalable smoothing, filtering, and data assimilation, via hierarchical sparse Cholesky, has become the standard for high-dimensional geostatistics and FFBS (Jurek et al., 2022, Jurek et al., 2020).
• Covariance/precision estimation in statistics: Penalized sparse Cholesky estimation is widely adopted in genomics, finance, and graphical modeling for full-rank, interpretable, and sparse positive definite covariance/precision estimation (Khare et al., 2016, Kang et al., 2017).
• Numerical linear algebra for PDEs: Rank-structured sparse Cholesky enables efficient direct solvers for large sparse SPD matrices, with near-linear storage and factorization times by compressing interactions in supernodes (Chadwick et al., 2015).
• Gaussian process regression and kernel methods: Sparse Cholesky via KL-minimization supports scalable GP inference, preconditioning, and active data selection (Schäfer et al., 2020, Huan et al., 2023).

Table: Representative Sparse Cholesky Applications

Area	Approach / Framework	Reference
Spatio-temporal filtering	Hierarchical Vecchia, sparse Cholesky	(Jurek et al., 2022, Jurek et al., 2020)
Large-scale kernel regression	KL-optimal sparse inverse Cholesky	(Schäfer et al., 2020, Huan et al., 2023)
Precision estimation in high-dim stats	Block Cholesky, MCD ensemble, ChoSelect	(Kang et al., 2023, Kang et al., 2018)
PDE solvers / SDD linear systems	RCHOL, sparsified Cholesky, rank-structured	(Chen et al., 2020, Lee et al., 2015, Chadwick et al., 2015)

7. Limitations, Practical Considerations, and Open Directions

Several factors influence the suitability and expected performance of sparse Cholesky approaches:

• Ordering and grouping: In cases where only partial ordering is available, block Cholesky or ensemble-based algorithms are preferable (Kang et al., 2023, Kang et al., 2018). Full-order methods may suffer in non-banded or dense graphs.
• Graph non-chordality/fill-in: For arbitrary graphs, exact zero-pattern preservation in $L$ requires augmentation to chordal or homogeneous supergraphs. Preprocessing with elimination trees, AMD, or nested dissection is standard for fill control (Khare et al., 2011, Lee et al., 2015).
• Computational trade-offs: Hierarchical and randomized algorithms can achieve linear or near-linear time and memory for massive problems, but parameter choices (conditioning set size, thresholding, aggregation) must balance statistical accuracy against cost (Schäfer et al., 2020, Jurek et al., 2022).
• Algorithmic developments: Potential improvements include obtaining $O(n/\epsilon)$ fill bounds for a given spectral error, designing purely combinatorial routines for sparsification, and extending techniques to general M–matrices and non-Gaussian models (Lee et al., 2015).

Sparse Cholesky decomposition thus constitutes a foundational tool, bridging numerical linear algebra, statistical estimation, and graphical modeling via principled factorization and computational strategies. Recent advances unify regression-based, penalized likelihood, graph-structured, and block-decomposition methodologies, allowing for scalable, interpretable, and theoretically justified inference in high dimensions and large-scale scientific computation.