Sparse Cholesky Factorization Research Articles

Avoiding breakdown in incomplete factorizations in low precision arithmetic

The emergence of low precision floating-point arithmetic in computer hardware has led to a resurgence of interest in the use of mixed precision numerical linear algebra. For linear systems of equations, there has been renewed enthusiasm for mixed precision variants of iterative refinement. We consider the iterative solution of large sparse systems using incomplete factorization preconditioners. The focus is on the robust computation of such preconditioners in half precision arithmetic and employing them to solve symmetric positive definite systems to higher precision accuracy; however, the proposed ideas can be applied more generally. Even for well-conditioned problems, incomplete factorizations can break down when small entries occur on the diagonal during the factorization. When using half precision arithmetic, overflows are an additional possible source of breakdown. We examine how breakdowns can be avoided and we implement our strategies within new half precision Fortran sparse incomplete Cholesky factorization software. Results are reported for a range of problems from practical applications. These demonstrate that, even for highly ill-conditioned problems, half precision preconditioners can potentially replace double precision preconditioners, although unsurprisingly this may be at the cost of additional iterations of a Krylov solver.

A mixed precision LOBPCG algorithm

The locally optimal block preconditioned conjugate gradient (LOBPCG) algorithm is a popular approach for computing a few smallest eigenvalues and the corresponding eigenvectors of a large Hermitian positive definite matrix A. In this work, we propose a mixed precision variant of LOBPCG that uses a (sparse) Cholesky factorization of A computed in reduced precision as the preconditioner. To further enhance performance, a mixed precision orthogonalization strategy is proposed. To analyze the impact of reducing precision in the preconditioner on performance, we carry out a rounding error and convergence analysis of PINVIT, a simplified variant of LOBPCG. Our theoretical results predict and our numerical experiments confirm that the impact on convergence remains marginal. In practice, our mixed precision LOBPCG algorithm typically reduces the computation time by a factor of 1.4--2.0 on both CPUs and GPUs.

Correlation-based sparse inverse Cholesky factorization for fast Gaussian-process inference

Gaussian processes are widely used as priors for unknown functions in statistics and machine learning. To achieve computationally feasible inference for large datasets, a popular approach is the Vecchia approximation, which is an ordered conditional approximation of the data vector that implies a sparse Cholesky factor of the precision matrix. The ordering and sparsity pattern are typically determined based on Euclidean distance of the inputs or locations corresponding to the data points. Here, we propose instead to use a correlation-based distance metric, which implicitly applies the Vecchia approximation in a suitable transformed input space. The correlation-based algorithm can be carried out in quasilinear time in the size of the dataset, and so it can be applied even for iterative inference on unknown parameters in the correlation structure. The correlation-based approach has two advantages for complex settings: It can result in more accurate approximations, and it offers a simple, automatic strategy that can be applied to any covariance, even when Euclidean distance is not applicable. We demonstrate these advantages in several settings, including anisotropic, nonstationary, multivariate, and spatio-temporal processes. We also illustrate our method on multivariate spatio-temporal temperature fields produced by a regional climate model.

Exactly Solving Sparse Rational Linear Systems via Roundoff-Error-Free Cholesky Factorizations

Exactly solving sparse symmetric positive definite (SPD) linear systems is a key problem in mathematics, engineering, and computer science. This paper derives two new sparse roundoff-error-free (REF) Cholesky factorization algorithms which exactly solve sparse SPD linear systems $A {x} = {b}$, where $A \in \mathbb{Q}^{n x n}$ and ${x}, {b} \in {Q}^{n x p}$. The key properties of these factorizations are that (1) they exclusively use integer-arithmetic and (2) in the bit-complexity model, they solve the linear system $A {x} = {b}$ in time proportional to the cost of the integer-arithmetic operations. Namely, the overhead related to data structures and ancillary operations (those not strictly required to perform the factorization) is subsumed by the cost of the integer-arithmetic operations that are essential/intrinsic to the factorization. Notably, to date our algorithms are the only exact algorithm for solving SPD linear systems with this asymptotically efficient complexity bound. Computationally, we show that the novel factorizations are faster than both sparse rational-arithmetic LDL and sparse exact LU factorization. Altogether, the derived sparse REF Cholesky factorizations present a framework to solve any rational SPD linear system exactly and efficiently.

Bayesian Nonstationary and Nonparametric Covariance Estimation for Large Spatial Data (with Discussion)

In spatial statistics, it is often assumed that the spatial field of interest is stationary and its covariance has a simple parametric form, but these assumptions are not appropriate in many applications. Given replicate observations of a Gaussian spatial field, we propose nonstationary and nonparametric Bayesian inference on the spatial dependence. Instead of estimating the quadratic (in the number of spatial locations) entries of the covariance matrix, the idea is to infer a near-linear number of nonzero entries in a sparse Cholesky factor of the precision matrix. Our prior assumptions are motivated by recent results on the exponential decay of the entries of this Cholesky factor for Matérn-type covariances under a specific ordering scheme. Our methods are highly scalable and parallelizable. We conduct numerical comparisons and apply our methodology to climate-model output, enabling statistical emulation of an expensive physical model.

Hierarchical sparse Cholesky decomposition with applications to high-dimensional spatio-temporal filtering

Spatial statistics often involves Cholesky decomposition of covariance matrices. To ensure scalability to high dimensions, several recent approximations have assumed a sparse Cholesky factor of the precision matrix. We propose a hierarchical Vecchia approximation, whose conditional-independence assumptions imply sparsity in the Cholesky factors of both the precision and the covariance matrix. This remarkable property is crucial for applications to high-dimensional spatiotemporal filtering. We present a fast and simple algorithm to compute our hierarchical Vecchia approximation, and we provide extensions to nonlinear data assimilation with non-Gaussian data based on the Laplace approximation. In several numerical comparisons, including a filtering analysis of satellite data, our methods strongly outperformed alternative approaches.

High-Dimensional Nonlinear Spatio-Temporal Filtering by Compressing Hierarchical Sparse Cholesky Factors

Spatio-temporal filtering is a common and challenging task in many environmental applications, where the evolution is often nonlinear and the dimension of the spatial state may be very high. We propose a scalable filtering approach based on a hierarchical sparse Cholesky representation of the filtering covariance matrix. At each time point, we compress the sparse Cholesky factor into a dense matrix with a small number of columns. After applying the evolution to each of these columns, we decompress to obtain a hierarchical sparse Cholesky factor of the forecast covariance, which can then be updated based on newly available data. We illustrate the Cholesky evolution via an equivalent representation in terms of spatial basis functions. We also demonstrate the advantage of our method in numerical comparisons, including using a high-dimensional and nonlinear Lorenz model.

Bregman primal\u2013dual first-order method and application to sparse semidefinite programming

We present a new variant of the Chambolle–Pock primal–dual algorithm with Bregman distances, analyze its convergence, and apply it to the centering problem in sparse semidefinite programming. The novelty in the method is a line search procedure for selecting suitable step sizes. The line search obviates the need for estimating the norm of the constraint matrix and the strong convexity constant of the Bregman kernel. As an application, we discuss the centering problem in large-scale semidefinite programming with sparse coefficient matrices. The logarithmic barrier function for the cone of positive semidefinite completable sparse matrices is used as the distance-generating kernel. For this distance, the complexity of evaluating the Bregman proximal operator is shown to be roughly proportional to the cost of a sparse Cholesky factorization. This is much cheaper than the standard proximal operator with Euclidean distances, which requires an eigenvalue decomposition.

Ensemble Kalman filter updates based on regularized sparse inverse Cholesky factors

AbstractThe ensemble Kalman filter (EnKF) is a popular technique for data assimilation in high-dimensional nonlinear state-space models. The EnKF represents distributions of interest by an ensemble, which is a form of dimension reduction that enables straightforward forecasting even for complicated and expensive evolution operators. However, the EnKF update step involves estimation of the forecast covariance matrix based on the (often small) ensemble, which requires regularization. Many existing regularization techniques rely on spatial localization, which may ignore long-range dependence. Instead, our proposed approach assumes a sparse Cholesky factor of the inverse covariance matrix, and the nonzero Cholesky entries are further regularized. The resulting method is highly flexible and computationally scalable. In our numerical experiments, our approach was more accurate and less sensitive to misspecification of tuning parameters than tapering-based localization.

Sparse Cholesky Factorization by Kullback--Leibler Minimization

We propose to compute a sparse approximate inverse Cholesky factor $L$ of a dense covariance matrix $\Theta$ by minimizing the Kullback--Leibler divergence between the Gaussian distributions $\mathcal{N}(0, \Theta)$ and $\mathcal{N}(0, L^{-\top} L^{-1})$, subject to a sparsity constraint. Surprisingly, this problem has a closed-form solution that can be computed efficiently, recovering the popular Vecchia approximation in spatial statistics. Based on recent results on the approximate sparsity of inverse Cholesky factors of $\Theta$ obtained from pairwise evaluation of Green's functions of elliptic boundary-value problems at points $\{x_{i}\}_{1 \leq i \leq N} \subset \mathbb{R}^{d}$, we propose an elimination ordering and sparsity pattern that allows us to compute $\epsilon$-approximate inverse Cholesky factors of such $\Theta$ in computational complexity $\mathcal{O}(N \log(N/\epsilon)^d)$ in space and $\mathcal{O}(N \log(N/\epsilon)^{2d})$ in time. To the best of our knowledge, this is the best asymptotic complexity for this class of problems. Furthermore, our method is embarrassingly parallel, automatically exploits low-dimensional structure in the data, and can perform Gaussian-process regression in linear (in $N$) space complexity. Motivated by its optimality properties, we propose applying our method to the joint covariance of training and prediction points in Gaussian-process regression, greatly improving stability and computational cost. Finally, we show how to apply our method to the important setting of Gaussian processes with additive noise, compromising neither accuracy nor computational complexity.

Accelerating Sparse Cholesky Factorization on Sunway Manycore Architecture

To improve the performance of sparse Cholesky factorization, existing research divides the adjacent columns of the sparse matrix with the same nonzero patterns into supernodes for parallelization. However, due to the various structures of sparse matrices, the computation of the generated supernodes varies significantly, and thus hard to optimize when computed by dense matrix kernels. Therefore, how to efficiently map sparse Choleksy factorization to the emerging architectures, such as Sunway many-core processor, remains an active research direction. In this article, we propose swCholesky , which is a highly optimized implementation of sparse Cholesky factorization on Sunway processor. Specifically, we design three kernel task queues and a dense matrix library to dynamically adapt to the kernel characteristics and architecture features. In addition, we propose an auto-tuning mechanism to search for the optimal settings of the important parameters in swCholesky . Our experiments show that swCholesky achieves better performance than state-of-the-art implementations.

Optimizing Regularized Cholesky Score for Order-Based Learning of Bayesian Networks.

Bayesian networks are a class of popular graphical models that encode causal and conditional independence relations among variables by directed acyclic graphs (DAGs). We propose a novel structure learning method, annealing on regularized Cholesky score (ARCS), to search over topological sorts, or permutations of nodes, for a high-scoring Bayesian network. Our scoring function is derived from regularizing Gaussian DAG likelihood, and its optimization gives an alternative formulation of the sparse Cholesky factorization problem from a statistical viewpoint. We combine simulated annealing over permutation space with a fast proximal gradient algorithm, operating on triangular matrices of edge coefficients, to compute the score of any permutation. Combined, the two approaches allow us to quickly and effectively search over the space of DAGs without the need to verify the acyclicity constraint or to enumerate possible parent sets given a candidate topological sort. The annealing aspect of the optimization is able to consistently improve the accuracy of DAGs learned by greedy and deterministic search algorithms. In addition, we develop several techniques to facilitate the structure learning, including pre-annealing data-driven tuning parameter selection and post-annealing constraint-based structure refinement. Through extensive numerical comparisons, we show that ARCS outperformed existing methods by a substantial margin, demonstrating its great advantage in structure learning of Bayesian networks from both observational and experimental data. We also establish the consistency of our scoring function in estimating topological sorts and DAG structures in the large-sample limit. Source code of ARCS is available at https://github.com/yeqiaoling/arcs_bn.

Sufficient and necessary conditions for solution finding in valuation-based systems

Valuation algebras abstract a large number of formalisms for automated reasoning and enable the definition of generic inference procedures. Many of these formalisms provide some notion of solution. Typical examples are satisfying assignments in constraint systems, models in logics or solutions to linear equation systems.Dynamic programming algorithms rely on low treewidth decompositions and can be understood as particular cases of a single algorithmic scheme for finding solutions in a valuation algebra. The most encompassing description of this algorithmic scheme to date has been proposed by Pouly and Kohlas together with sufficient conditions for its correctness. Unfortunately, the formalization relies on a theorem for which we provide a counterexample. In spite of that, the mainline of Pouly and Kohlas' theory is correct, although some of the necessary conditions have to be revised. In this paper we analyze the impact that the counterexample has on the theory, and rebuild the theory providing correct sufficient conditions for the algorithms. Furthermore, we also provide necessary conditions for the algorithms, allowing for a sharper characterization of when the algorithmic scheme can be applied.The contribution of the paper is not limited to correcting Pouly and Kohlas' theory. First, we generalize their results from discrete tuples to tuple systems. This allows us to cover some classical algorithms, such as sparse Cholesky factorization, as well as more recently proposed algorithms, such as Snowball (sparse all-pairs shortest path problems), as particular cases of the generic valuation-based algorithms presented here.For the particular case of valuation algebras induced by selective semirings, Pouly and Kohlas presented an algorithm for finding all solutions without requiring any additional condition for correctness. We characterize a necessary and a sufficient condition for it. Furthermore, we also introduce a new algorithm that requires no necessary condition.

Estimating variance components in population scale family trees.

The rapid digitization of genealogical and medical records enables the assembly of extremely large pedigree records spanning millions of individuals and trillions of pairs of relatives. Such pedigrees provide the opportunity to investigate the sociological and epidemiological history of human populations in scales much larger than previously possible. Linear mixed models (LMMs) are routinely used to analyze extremely large animal and plant pedigrees for the purposes of selective breeding. However, LMMs have not been previously applied to analyze population-scale human family trees. Here, we present Sparse Cholesky factorIzation LMM (Sci-LMM), a modeling framework for studying population-scale family trees that combines techniques from the animal and plant breeding literature and from human genetics literature. The proposed framework can construct a matrix of relationships between trillions of pairs of individuals and fit the corresponding LMM in several hours. We demonstrate the capabilities of Sci-LMM via simulation studies and by estimating the heritability of longevity and of reproductive fitness (quantified via number of children) in a large pedigree spanning millions of individuals and over five centuries of human history. Sci-LMM provides a unified framework for investigating the epidemiological history of human populations via genealogical records.

Linear-Time Algorithm for Learning Large-Scale Sparse Graphical Models

We consider the graphical lasso, a popular optimization problem for learning the sparse representations of high-dimensional datasets, which is well-known to be computationally expensive for large-scale problems. A recent line of results has shown–under mild assumptions–that the sparsity pattern of the graphical lasso estimator can be retrieved by soft-thresholding the sample covariance matrix. Based on this result, a closed-form solution has been obtained that is optimal when the thresholded sample covariance matrix has an acyclic structure. In this paper, we prove an extension of this result to generalized graphical lasso (GGL), where additional sparsity constraints are imposed based on prior knowledge. Furthermore, we describe a recursive closed-form solution for the problem when the thresholded sample covariance matrix is chordal. By building upon this result, we describe a novel Newton-Conjugate Gradient algorithm that can efficiently solve the GGL with general structures. Assuming that the thresholded sample covariance matrix is sparse with a sparse Cholesky factorization, we prove that the algorithm converges to an $\epsilon $ -accurate solution in $O(n\log (1/\epsilon))$ time and $O(n)$ memory. The algorithm is highly efficient in practice: we solve instances with as many as 200000 variables to 7–9 digits of accuracy in less than an hour on a standard laptop computer running MATLAB.

An MHSS-like iteration method for two-by-two linear systems with application to FDE optimization problems

In this paper, we exploit the numerical solvers for a fractional differential equations (FDE) optimization problem with constraints given by fractional elliptic state equations. The discretized linear system can be rewritten as a two-by-two linear system. By making use of the strategy of modified Hermitian and skew-Hermitian splitting (MHSS) iteration method proposed by Bai, we propose an MHSS-like iteration method. Comparing with the MHSS iteration method, the MHSS-like method only requires one to solve the two-by-two linear systems by a sparse Cholesky factorization and a fast Fourier transform. Hence, the MHSS-like method has less workload than the MHSS method. The convergence properties and the quasi-optimal parameters are analyzed in detail. Numerical examples are used to testify the efficiency of the new method.

GenoGAM 2.0: scalable and efficient implementation of genome-wide generalized additive models for gigabase-scale genomes

BackgroundGenoGAM (Genome-wide generalized additive models) is a powerful statistical modeling tool for the analysis of ChIP-Seq data with flexible factorial design experiments. However large runtime and memory requirements of its current implementation prohibit its application to gigabase-scale genomes such as mammalian genomes.ResultsHere we present GenoGAM 2.0, a scalable and efficient implementation that is 2 to 3 orders of magnitude faster than the previous version. This is achieved by exploiting the sparsity of the model using the SuperLU direct solver for parameter fitting, and sparse Cholesky factorization together with the sparse inverse subset algorithm for computing standard errors. Furthermore the HDF5 library is employed to store data efficiently on hard drive, reducing memory footprint while keeping I/O low. Whole-genome fits for human ChIP-seq datasets (ca. 300 million parameters) could be obtained in less than 9 hours on a standard 60-core server. GenoGAM 2.0 is implemented as an open source R package and currently available on GitHub. A Bioconductor release of the new version is in preparation.ConclusionsWe have vastly improved the performance of the GenoGAM framework, opening up its application to all types of organisms. Moreover, our algorithmic improvements for fitting large GAMs could be of interest to the statistical community beyond the genomics field.

Optimized sparse Cholesky factorization on hybrid multicore architectures

We present techniques for supernodal sparse Cholesky factorization on a hybrid multicore platform consisting of a multicore CPU and GPU. The techniques are the subtree algorithm, pipelining and multithreading. The subtree algorithm [15] minimizes PCIe transmissions by storing an entire branch of the elimination tree in the GPU memory (the elimination tree is a tree data structure describing the workflow of the factorization), and also reduces the total kernel launch time by launching BLAS kernels in batches. The pipelining technique overlaps the execution of GPU kernels and PCIe data transfers. The multithreading technique [17] creates multiple threads for both the CPU and the GPU, to utilize concurrency of the elimination tree. Our experimental results on a platform consisting of an Intel multicore processor along with an Nvidia GPU indicate a significant improvement in performance and energy over CHOLMOD (SuiteSparse 4.5.3), a sparse algorithm, after these techniques are applied.

Fast learning of scale‐free networks based on Cholesky factorization

Recovering network connectivity structure from high-dimensional observations is of increasing importance in statistical learning applications. A prominent approach is to learn a Sparse Gaussian Markov Random Field by optimizing regularized maximum likelihood, where the sparsity is induced by imposing L1 norm on the entries of a precision matrix. In this article, we shed light on an alternative objective, where instead of precision, its Cholesky factor is penalized by the L1 norm. We show that such an objective criterion possesses attractive properties that allowed us to develop a very fast Scale-Free Networks Estimation Through Cholesky factorization (SNETCH) optimization algorithm based on coordinate descent, which is highly parallelizable and can exploit an active set approach. The approach is particularly suited for problems with structures that allow sparse Cholesky factor, an important example being scale-free networks. Evaluation on synthetically generated examples and high-impact applications from a biomedical domain of up to more than 900,000 variables provides evidence that for such tasks the SNETCH algorithm can learn the underlying structure more accurately, and an order of magnitude faster than state-of-the-art approaches based on the L1 penalized precision matrix.

A Hybrid CPU-GPU Multifrontal Optimizing Method in Sparse Cholesky Factorization

In many scientific computing applications, sparse Cholesky factorization is used to solve large sparse linear equations in distributed environment. GPU computing is a new way to solve the problem. However, sparse Cholesky factorization on GPU is hardly to achieve excellent performance due to the structure irregularity of matrix and the low GPU resource utilization. A hybrid CPU-GPU implementation of sparse Cholesky factorization is proposed based on multifrontal method. A large sparse coefficient matrix is decomposed into a series of small dense matrices (frontal matrices) in the method, and then multiple GEMM (General Matrix-matrix Multiplication) operations are computed on them. GEMMs are the main operations in sparse Cholesky factorization, but they are hardly to perform better in parallel on GPU. In order to improve the performance, the scheme of multiple task queues is adopted to perform multiple GEMMs parallelized with multifrontal method; all GEMM tasks are scheduled dynamically on GPU and CPU based on computation scales for load balance and computing-time reduction. Experimental results show that the approach can outperform the implementations of cuBLAS, achieving up to 1.98× speedup on GTX460 (Fermi micro-architecture) and 3.06× speedup on K20m (Kepler micro-architecture), respectively.