arXiv subjects

Bongjung Sung

Publications and source records attributed to Bongjung Sung.

3 recordsLinked to original sources

arXiv2026

Testing Separability of High-Dimensional Covariance Matrices

Due to their parsimony, separable covariance models have been popular in modeling matrix-variate data. However, the inference from such a model may be misleading if the population covariance matrix $Σ$ is actually non-separable, motivating the use of statistical tests of separability. The existing separability tests suffer mainly from two issues: 1) test statistics that are not well-defined in high-dimensional settings, 2) low power for small sample sizes and null distributions that depend on unknown parameters, preventing exact error rate control. To address these issues, we propose novel invariant tests using the core covariance matrix, a complementary notion to a separable covariance matrix. We show that testing separability of $Σ$ is equivalent to testing sphericity of its core component. With this insight, we construct test statistics that are well-defined in high-dimensional settings and have distributions that are invariant under the null hypothesis of separability, allowing for exact simulation of null distributions. We establish the asymptotic properties of some test statistics by proving the asymptotic spectral equivalence between the sample covariance matrix and its core in a $p/n\rightarrowγ\in(0,\infty)$ regime. The large power of our proposed tests relative to existing procedures is demonstrated numerically.

math.ST↗

arXiv2026

Covariance Estimation for Matrix-variate Data via Fixed-rank Core Covariance Geometry

We study the geometry of the fixed-rank core covariance manifold arising from the Kronecker-core decomposition of covariance matrices. As shown in Hoff, McCormack, and Zhang (2023), every covariance matrix $Σ$ of $p_1\times p_2$ matrix-variate data uniquely decomposes into a separable component $K$ and a core component $C$. Such a decomposition also exists for rank-$r$ $Σ$ if $p_1/p_2+p_2/p_1<r$, with $C$ sharing the same rank. If this core $C$ exhibits a partial-isotropy structure, then a partial-isotropy rank-$r$ core is a non-trivial convex combination of a rank-$r$ core and $I_p$ for $p:=p_1p_2$, where the weight on $I_p$ measures the deviation of $Σ$ from separability. This motivates studying the geometry of the space of rank-$r$ cores, $\mathcal{C}_{p_1,p_2,r}^+$. We show that $\mathcal{C}_{p_1,p_2,r}^+$ is a smooth manifold, except for a measure-zero subset associated with canonical decomposability. When $r=p$, $\mathcal{C}_{p_1,p_2}^{++}:=\mathcal{C}_{p_1,p_2,p}^+$ is itself a smooth manifold. The geometric properties, including smoothness of the positive definite cone via separability and the Riemannian gradient and Hessian operator relevant to $\mathcal{C}_{p_1,p_2,r}^+$, are also derived. As an application, we propose a partial-isotropy core shrinkage estimator for matrix-variate data.

math.DG↗

arXiv2021

Covariance Structure Estimation with Laplace Approximation

Gaussian covariance graph model is a popular model in revealing underlying dependency structures among random variables. A Bayesian approach to the estimation of covariance structures uses priors that force zeros on some off-diagonal entries of covariance matrices and put a positive definite constraint on matrices. In this paper, we consider a spike and slab prior on off-diagonal entries, which uses a mixture of point-mass and normal distribution. The point-mass naturally introduces sparsity to covariance structures so that the resulting posterior from this prior renders covariance structure learning. Under this prior, we calculate posterior model probabilities of covariance structures using Laplace approximation. We show that the error due to Laplace approximation becomes asymptotically marginal at some rate depending on the posterior convergence rate of covariance matrix under the Frobenius norm. With the approximated posterior model probabilities, we propose a new framework for estimating a covariance structure. Since the Laplace approximation is done around the mode of conditional posterior of covariance matrix, which cannot be obtained in the closed form, we propose a block coordinate descent algorithm to find the mode and show that the covariance matrix can be estimated using this algorithm once the structure is chosen. Through a simulation study based on five numerical models, we show that the proposed method outperforms graphical lasso and sample covariance matrix in terms of root mean squared error, max norm, spectral norm, specificity, and sensitivity. Also, the advantage of the proposed method is demonstrated in terms of accuracy compared to our competitors when it is applied to linear discriminant analysis (LDA) classification to breast cancer diagnostic dataset.

stat.ME↗