A novel convex optimization framework in which an initial matrix is projected onto an elliptope under a positive semidefiniteness constraint, providing a principled and tunable way to construct correlation matrices suitable for benchmarking statistical methods for graphical model inference.
Abstract
This work addresses the generation of theoretical correlation matrices with prescribed sparsity patterns associated to graph structures. We propose a novel convex optimization framework in which an initial matrix is projected onto an elliptope under a positive semidefiniteness constraint. Several numerical schemes are implemented and compared. The problem falls within the broader class of matrix completion, where off-diagonal entries corresponding to absent edges are fixed to zero and diagonal entries are fixed to one. Beyond this structural constraint, the approach offers greater flexibility than existing methods by allowing control over the mean of the off-diagonal entry distribution, enabling the generation of correlation matrices that better reflect realistic data. This procedure is not designed to yield a uniform distribution over the feasible set; rather, it provides a principled and tunable way to construct correlation matrices suitable for benchmarking statistical methods for graphical model inference. Theoretical guarantees on the existence of solutions are established, both in the general setting and under the additional mean constraint. Simulation studies illustrate the properties of the generated matrices with respect to graph structure. The methodology is applied to two real-world datasets from neuroscience and finance, and a comparison with GAN-based correlation matrix generation is provided.
A Derivative-Free Optimization (DFO) model is developed and analyzed for solving inverse structured symmetric matrix problems for which the eigenvalues are specified. Some (zero and nonzero) entries are preassigned and cannot be changed, while others should be nonzero but their values are not given. The rest of the entries are completely free. The obtained matrix must meet these criteria and have the specified eigenvalues. This specialized inverse eigenvalue problem is relevant to various applications and is linked to determining the graph, with weights on the undirected edges, of the matrix associated with its sparse pattern. Our novel optimization model requires computing the eigenvalues of a symmetric matrix to evaluate the non-differentiable objective function. We apply deterministic DFO schemes, specifically the global variant GLODS of the well-known family of directional direct search (DDS) methods. We discuss its convergence properties which are based on the fact that the objective function of our model is Lipschitz continuous. Additionally, we explore the potential benefits of using several well-established heuristic strategies to solve the proposed optimization model. We present preliminary numerical results to illustrate and compare the performance of the considered deterministic and heuristic DFO options in various possible scenarios.
Carmo P. Brás, E.H.M. Krulikovski, M. Raydan· 0 citations
Estimating multiple precision matrices in high-dimension presents significant challenges, particularly when distinct datasets share a common conditional dependency structure but exhibit population-specific interaction strengths. We address this problem by introducing the Multiplicative Graphical Lasso (Mglasso), a method for jointly estimating precision matrices across multiple Gaussian graphical models under a shared sparsity constraint. Each precision matrix is decomposed as a Schur-Hadamard product of a shared structural matrix $\boldsymbol{\Theta}$, which encodes the common conditional independence graph, and a population-specific matrix $\boldsymbol{\Gamma}_{l}$, which captures variation in edge strengths across populations. We optimize a penalized log-likelihood that utilizes an $\ell_1$-penalty to enforce common sparsity and a Frobenius norm penalty to regulate population-specific variations. The optimization is efficiently performed using the Alternating Direction Method of Multipliers (ADMM) algorithm integrated with gradient descent. Theoretically, we establish the local strict convexity of the objective function and provide rigorous high-dimensional consistency guarantees, including supremum norm error bounds and exact support recovery under sub-Gaussian tail conditions. Extensive simulations show superior model selection consistency at smaller sample sizes compared to the benchmark Group Graphical Lasso (GGL). Finally, the method's practical utility is further validated through real-world applications.
S. Bhowal, Debashis Paul, Gopal K. Basak et al.· 0 citations
Abstract.
This paper gives a necessary and sufficient condition for a nonnegative matrix that has an orthogonal nonnegative matrix factorization (ONMF) via characterization of the null space. We propose an optimization model to minimize the Frobenius norm of the product of a given nonnegative matrix and a variable matrix subject to the constraints defined by the necessary and sufficient condition. Moreover, we present an augmented Lagrangian algorithm for solving this minimization model and prove the global convergence to a stationary point. Two factor matrices for the ONMF of the given matrix can be easily obtained by the outputs of the algorithm. Preliminary numerical results using synthetic and real-world data with applications in clustering show that our approach outperforms some existing ONMF methods regarding accuracy and robustness.
Reproducibility of computational results. This paper has been awarded the “SIAM Reproducibility Badge: Code and data available” as recognition that the authors have followed reproducibility principles valued by SIMAX and the scientific computing community. Code and data that allow readers to reproduce the results in this paper are available at https://github.com/Qilun-Luo/ONMF . [Formula: see text]
Specifying a prior over the space of correlation matrices is a persistent challenge in Bayesian analysis. The space is a curved manifold whose dimension grows quadratically with the number of variables, making substantive prior beliefs difficult to encode.\\ We propose a distance-based prior that assigns mass decaying exponentially in the Fisher arc-length distance from a user-specified reference correlation matrix, enabling shrinkage toward any target correlation structure rather than being confined to the identity matrix. Formally, this is constructed as a Penalised Complexity prior, but its interpretation shifts accordingly: unless the chosen target represents a structurally simpler state, the shrinkage penalises deviation rather than complexity in the usual sense. To accommodate conditional independence constraints, we introduce a parameterisation that constructs the correlation matrix via the Cholesky factor of the inverse correlation matrix with respect to a user-supplied graph, thereby reducing the number of free parameters from one per variable pair to one per graph edge. The prior is proper for every positive value of its rate parameter, accommodates correlations of either sign under any graph structure, and reduces to a fully unstructured prior when the graph is complete. A direct sampling algorithm is provided, enabling prior predictive checks and sensitivity analysis, implemented within the \texttt{graphpcor} package.
A. Freni-Sterrantino, J. V. Niekerk, E. Krainski et al.· 1 citation
Matrix factorisation is a fundamental tool for exploiting low-dimensional structure in high-dimensional data, with applications such as data compression, denoising, structure discovery, interpretable representation learning, and dimensionality reduction. Compared to conventional two-factor models, matrix tri-factorisation provides greater modelling flexibility, while sparsity constraints often improve both interpretability and recovery performance. Although the role of sparsity has been extensively studied for two-factor matrix factorisation, rigorous theoretical guarantees for general real-valued matrix tri-factorisation remain largely unexplored. To address this gap, we establish, to the best of our knowledge, the first rigorous theoretical study for sparsity-induced identifiability in general real-valued matrix tri-factorisation. Our analysis is enabled by a novel decomposition strategy that transforms the original problem into two coupled auxiliary factorisation problems, while preserving the structural information necessary to the recovery of the original factor matrices from the observations. Building upon this decomposition, we derive recovery guarantees and structural consistency results that characterise how coefficient sparsity influences the sufficient recovery conditions, convergence behaviour, spectral approximation error, high-probability bounds, and structure preservation. Comprehensive Monte Carlo experiments validate the proposed theory and demonstrate close agreement between the theoretical results and empirical observations.
Over-parameterized linear regression has been widely studied over the last decade. However, most existing works assume that the covariates are independent and that their covariance matrices are non-degenerate. In this paper, we relax both assumptions and derive deterministic equivalents for the prediction risk in a vanishing-ridge regime. We show that degeneracy of the covariance matrices and dependence can lead to multiple descent, and characterize where the corresponding peaks can occur. Our proofs use a novel graph representation of the variance profile. We show that maximum matchings and the Dulmage--Mendelsohn decomposition of the associated bipartite graph identify the configurations at which the variance becomes singular.
Kevin Han Huang, Haoyu Ye, Somak Laha et al.· 0 citations