arXiv subjects

Debashis Paul

Publications and source records attributed to Debashis Paul.

At least 19 recordsLinked to original sources

arXiv2026

Estimation of multiple precision matrices under shared support with heterogeneous edge strengths

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.

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arXiv2025

Statistical inference using debiased group graphical lasso for multiple sparse precision matrices

Debiasing group graphical lasso estimates enables statistical inference when multiple Gaussian graphical models share a common sparsity pattern. We analyze the estimation properties of group graphical lasso, establishing convergence rates and model selection consistency under irrepresentability conditions. Based on these results, we construct debiased estimators that are asymptotically Gaussian, allowing hypothesis testing for linear combinations of precision matrix entries across populations. We also investigate regimes where irrepresentibility conditions does not hold, showing that consistency can still be attained in moderately high-dimensional settings. Simulation studies confirm the theoretical results, and applications to real datasets demonstrate the practical utility of the method.

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arXiv2025

LSD of sample covariances of superposition of matrices with separable covariance structure

We study the asymptotic behavior of the spectra of matrices of the form $S_n = \frac{1}{n}XX^*$ where $X =\sum_{r=1}^K X_r$, where $X_r = A_r^\frac{1}{2}Z_rB_r^\frac{1}{2}$, $K \in \mathbb{N}$ and $A_r,B_r$ are sequences of positive semi-definite matrices of dimensions $p\times p$ and $n\times n$, respectively. We establish the existence of a limiting spectral distribution for $S_n$ by assuming that matrices $\{A_r\}_{r=1}^K$ are simultaneously diagonalizable and $\{B_r\}_{r=1}^K$ are simultaneously digaonalizable, and that the joint spectral distributions of $\{A_r\}_{r=1}^K$ and $\{B_r\}_{r=1}^K$ converge to $K$-dimensional distributions, as $p,n\to \infty$ such that $p/n \to c \in (0,\infty)$. The LSD of $S_n$ is characterized by system of equations with unique solutions within the class of Stieltjes transforms of measures on $\mathbb{R}_+$. These results generalize existing results on the LSD of sample covariances when the data matrices have a separable covariance structure.

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arXiv2025

Spectral estimation for high-dimensional linear processes

We propose a novel estimation procedure for certain spectral distributions associated with a class of high dimensional linear time series. The processes under consideration are of the form $X_t = \sum_{\ell=0}^\infty \mathbf{A}_\ell Z_{t-\ell}$ with iid innovations $(Z_t)$. The key structural assumption is that the coefficient matrices and the variance of the innovations are simultaneously diagonalizable in a common orthonormal basis. We develop a strategy for estimating the joint spectral distribution of the coefficient matrices and the innovation variance by making use of the asymptotic behavior of the eigenvalues of appropriately weighted integrals of the sample periodogram. Throughout we work under the asymptotic regime $p,n \to \infty$, such that $p/n\to c \in (0,\infty)$, where $p$ is the dimension and $n$ is the sample size. Under this setting, we first establish a weak limit for the empirical distribution of eigenvalues of the aforementioned integrated sample periodograms. This result is proved using techniques from random matrix theory, in particular the characterization of weak convergence by means of the Stieltjes transform of relevant distributions. We utilize this result to develop an estimator of the joint spectral distribution of the coefficient matrices, by minimizing an $L^\kappa$ discrepancy measure, for $\kappa \geq 1$, between the empirical and limiting Stieltjes transforms of the integrated sample periodograms. This is accomplished by assuming that the joint spectral distribution is a discrete mixture of point masses. We also prove consistency of the estimator corresponding to the $L^2$ discrepancy measure. We illustrate the methodology through simulations and an application to stock price data from the S\&P 500 series.

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arXiv2025

LSD of the Commutator of two data Matrices

We study the spectral properties of a class of random matrices of the form $S_n^{-} = n^{-1}(X_1 X_2^* - X_2 X_1^*)$ where $X_k = \Sigma_k^{1/2}Z_k$, $Z_k$'s are independent $p\times n$ complex-valued random matrices, and $\Sigma_k$ are $p\times p$ positive semi-definite matrices that commute and are independent of the $Z_k$'s for $k=1,2$. We assume that $Z_k$'s have independent entries with zero mean and unit variance. The skew-symmetric/skew-Hermitian matrix $S_n^{-}$ will be referred to as a random commutator matrix associated with the samples $X_1$ and $X_2$. We show that, when the dimension $p$ and sample size $n$ increase simultaneously, so that $p/n \to c \in (0,\infty)$, there exists a limiting spectral distribution (LSD) for $S_n^{-}$, supported on the imaginary axis, under the assumptions that the joint spectral distribution of $\Sigma_1, \Sigma_2$ converges weakly and the entries of $Z_k$'s have moments of sufficiently high order. This nonrandom LSD can be described through its Stieltjes transform, which satisfies a system of Mar\v{c}enko-Pastur-type functional equations. Moreover, we show that the companion matrix $S_n^{+} = n^{-1}(X_1X_2^* + X_2X_1^*)$, under identical assumptions, has an LSD supported on the real line, which can be similarly characterized.

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arXiv2024

Inferring Latent Graphs from Stationary Signals Using a Graphical Autoregressive Model

Graphs are an intuitive way to represent relationships between variables in fields such as finance and neuroscience. However, these graphs often need to be inferred from data. In this paper, we propose a novel framework to infer a latent graph by treating the observed multidimensional data as graph-referenced stationary signals. Specifically, we introduce the graphical autoregressive model (GAR), where the inverse covariance matrix of the observed signals is expressed as a second-order polynomial of the normalized graph Laplacian of the latent graph. The GAR model extends the autoregressive model from time series analysis to general undirected graphs, offering a new approach to graph inference. To estimate the latent graph, we develop a three-step procedure based on penalized maximum likelihood, supported by theoretical analysis and numerical experiments. Simulation studies and an application to S&P 500 stock price data show that the GAR model can outperform Gaussian graphical models when it fits the observed data well. Our results suggest that the GAR model offers a promising new direction for inferring latent graphs across diverse applications. Codes and example scripts are available at https://github.com/jed-harwood/SGM .

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arXiv2024

Limiting Spectral Distribution of a Random Commutator Matrix

We study the spectral properties of a class of random matrices of the form $S_n^{-} = n^{-1}(X_1 X_2^* - X_2 X_1^*)$ where $X_k = \Sigma^{1/2}Z_k$, for $k=1,2$, $Z_k$'s are independent $p\times n$ complex-valued random matrices, and $\Sigma$ is a $p\times p$ positive semi-definite matrix, independent of the $Z_k$'s. We assume that $Z_k$'s have independent entries with zero mean and unit variance. The skew-symmetric/skew-Hermitian matrix $S_n^{-}$ will be referred to as a random commutator matrix associated with the samples $X_1$ and $X_2$. We show that, when the dimension $p$ and sample size $n$ increase simultaneously, so that $p/n \to c \in (0,\infty)$, there exists a limiting spectral distribution (LSD) for $S_n^{-}$, supported on the imaginary axis, under the assumptions that the spectral distribution of $\Sigma$ converges weakly and the entries of $Z_k$'s have moments of sufficiently high order. This nonrandom LSD can be described through its Stieltjes transform, which satisfies a coupled Mar\v{c}enko-Pastur-type functional equations. In the special case when $\Sigma = I_p$, we show that the LSD of $S_n^{-}$ is a mixture of a degenerate distribution at zero (with positive mass if $c > 2$), and a continuous distribution with a symmetric density function supported on a compact interval on the imaginary axis. Moreover, we show that the companion matrix $S_n^{+} = \Sigma_n^\frac{1}{2}(Z_1Z_2^* + Z_2Z_1^*)\Sigma_n^\frac{1}{2}$, under identical assumptions, has an LSD supported on the real line, which can be similarly characterized.

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arXiv2024

Detecting Spectral Breaks in Spiked Covariance Models

In this paper, the key objects of interest are the sequential covariance matrices $\mathbf{S}_{n,t}$ and their largest eigenvalues. Here, the matrix $\mathbf{S}_{n,t}$ is computed as the empirical covariance associated with observations $\{\mathbf{x}_1,\ldots,\mathbf{x}_{ \lfloor nt \rfloor } \}$, for $t\in [0,1]$. The observations $\mathbf{x}_1,\ldots,\mathbf{x}_n$ are assumed to be i.i.d. $p$-dimensional vectors with zero mean, and a covariance matrix that is a fixed-rank perturbation of the identity matrix. Treating $\{ \mathbf{S}_{n,t}\}_{t \in [0,1]}$ as a matrix-valued stochastic process indexed by $t$, we study the behavior of the largest eigenvalues of $\mathbf{S}_{n,t}$, as $t$ varies, with $n$ and $p$ increasing simultaneously, so that $p/n \to y \in (0,1)$. As a key contribution of this work, we establish the weak convergence of the stochastic process corresponding to the sample spiked eigenvalues, if their population counterparts exceed the critical phase-transition threshold. Our analysis of the limiting process is fully comprehensive revealing, in general, non-Gaussian limiting processes. As an application, we consider a class of change-point problems, where the interest is in detecting structural breaks in the covariance caused by a change in magnitude of the spiked eigenvalues. For this purpose, we propose two different maximal statistics corresponding to centered spiked eigenvalues of the sequential covariances. We show the existence of limiting null distributions for these statistics, and prove consistency of the test under fixed alternatives. Moreover, we compare the behavior of the proposed tests through a simulation study.

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arXiv2024

Meta-Learning with Generalized Ridge Regression: High-dimensional Asymptotics, Optimality and Hyper-covariance Estimation

Meta-learning involves training models on a variety of training tasks in a way that enables them to generalize well on new, unseen test tasks. In this work, we consider meta-learning within the framework of high-dimensional multivariate random-effects linear models and study generalized ridge-regression based predictions. The statistical intuition of using generalized ridge regression in this setting is that the covariance structure of the random regression coefficients could be leveraged to make better predictions on new tasks. Accordingly, we first characterize the precise asymptotic behavior of the predictive risk for a new test task when the data dimension grows proportionally to the number of samples per task. We next show that this predictive risk is optimal when the weight matrix in generalized ridge regression is chosen to be the inverse of the covariance matrix of random coefficients. Finally, we propose and analyze an estimator of the inverse covariance matrix of random regression coefficients based on data from the training tasks. As opposed to intractable MLE-type estimators, the proposed estimators could be computed efficiently as they could be obtained by solving (global) geodesically-convex optimization problems. Our analysis and methodology use tools from random matrix theory and Riemannian optimization. Simulation results demonstrate the improved generalization performance of the proposed method on new unseen test tasks within the considered framework.

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arXiv2021

Variance Estimation and Confidence Intervals from High-dimensional Genome-wide Association Studies Through Misspecified Mixed Model Analysis

We study variance estimation and associated confidence intervals for parameters characterizing genetic effects from genome-wide association studies (GWAS) misspecified mixed model analysis. Previous studies have shown that, in spite of the model misspecification, certain quantities of genetic interests are estimable, and consistent estimators of these quantities can be obtained using the restricted maximum likelihood (REML) method under a misspecified linear mixed model. However, the asymptotic variance of such a REML estimator is complicated and not ready to be implemented for practical use. In this paper, we develop practical and computationally convenient methods for estimating such asymptotic variances and constructing the associated confidence intervals. Performance of the proposed methods is evaluated empirically based on Monte-Carlo simulations and real-data application.

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arXiv2020

Estimating Fiber Orientation Distribution through Blockwise Adaptive Thresholding with Application to HCP Young Adults Data

Due to recent technological advances, large brain imaging data sets can now be collected. Such data are highly complex so extraction of meaningful information from them remains challenging. Thus, there is an urgent need for statistical procedures that are computationally scalable and can provide accurate estimates that capture the neuronal structures and their functionalities. We propose a fast method for estimating the fiber orientation distribution(FOD) based on diffusion MRI data. This method models the observed dMRI signal at any voxel as a convolved and noisy version of the underlying FOD, and utilizes the spherical harmonics basis for representing the FOD, where the spherical harmonic coefficients are adaptively and nonlinearly shrunk by using a James-Stein type estimator. To further improve the estimation accuracy by enhancing the localized peaks of the FOD, as a second step a super-resolution sharpening process is then applied. The resulting estimated FODs can be fed to a fiber tracking algorithm to reconstruct the white matter fiber tracts. We illustrate the overall methodology using both synthetic data and data from the Human Connectome Project.

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arXiv2019

Sparse Equisigned PCA: Algorithms and Performance Bounds in the Noisy Rank-1 Setting

Singular value decomposition (SVD) based principal component analysis (PCA) breaks down in the high-dimensional and limited sample size regime below a certain critical eigen-SNR that depends on the dimensionality of the system and the number of samples. Below this critical eigen-SNR, the estimates returned by the SVD are asymptotically uncorrelated with the latent principal components. We consider a setting where the left singular vector of the underlying rank one signal matrix is assumed to be sparse and the right singular vector is assumed to be equisigned, that is, having either only nonnegative or only nonpositive entries. We consider six different algorithms for estimating the sparse principal component based on different statistical criteria and prove that by exploiting sparsity, we recover consistent estimates in the low eigen-SNR regime where the SVD fails. Our analysis reveals conditions under which a coordinate selection scheme based on a \textit{sum-type decision statistic} outperforms schemes that utilize the $\ell_1$ and $\ell_2$ norm-based statistics. We derive lower bounds on the size of detectable coordinates of the principal left singular vector and utilize these lower bounds to derive lower bounds on the worst-case risk. Finally, we verify our findings with numerical simulations and illustrate the performance with a video data example, where the interest is in identifying objects.

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arXiv2018

High-dimensional general linear hypothesis tests via non-linear spectral shrinkage

We are interested in testing general linear hypotheses in a high-dimensional multivariate linear regression model. The framework includes many well-studied problems such as two-sample tests for equality of population means, MANOVA and others as special cases. A family of rotation-invariant tests is proposed that involves a flexible spectral shrinkage scheme applied to the sample error covariance matrix. The asymptotic normality of the test statistic under the null hypothesis is derived in the setting where dimensionality is comparable to sample sizes, assuming the existence of certain moments for the observations. The asymptotic power of the proposed test is studied under various local alternatives. The power characteristics are then utilized to propose a data-driven selection of the spectral shrinkage function. As an illustration of the general theory, we construct a family of tests involving ridge-type regularization and suggest possible extensions to more complex regularizers. A simulation study is carried out to examine the numerical performance of the proposed tests.

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arXiv2017

A Multi-Resolution Model for Non-Gaussian Random Fields on a Sphere with Application to Ionospheric Electrostatic Potentials

Gaussian random fields have been one of the most popular tools for analyzing spatial data. However, many geophysical and environmental processes often display non-Gaussian characteristics. In this paper, we propose a new class of spatial models for non-Gaussian random fields on a sphere based on a multi-resolution analysis. Using a special wavelet frame, named spherical needlets, as building blocks, the proposed model is constructed in the form of a sparse random effects model. The spatial localization of needlets, together with carefully chosen random coefficients, ensure the model to be non-Gaussian and isotropic. The model can also be expanded to include a spatially varying variance profile. The special formulation of the model enables us to develop efficient estimation and prediction procedures, in which an adaptive MCMC algorithm is used. We investigate the accuracy of parameter estimation of the proposed model, and compare its predictive performance with that of two Gaussian models by extensive numerical experiments. Practical utility of the proposed model is demonstrated through an application of the methodology to a data set of high-latitude ionospheric electrostatic potentials, generated from the LFM-MIX model of the magnetosphere-ionosphere system.

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arXiv2016

Modeling Tangential Vector Fields on a Sphere

Physical processes that manifest as tangential vector fields on a sphere are common in geophysical and environmental sciences. These naturally occurring vector fields are often subject to physical constraints, such as being curl-free or divergence-free. We construct a new class of parametric models for cross-covariance functions of curl-free and divergence-free vector fields that are tangential to the unit sphere. These models are constructed by applying the surface gradient or the surface curl operator to scalar random potential fields defined on the unit sphere. We propose a likelihood-based estimation procedure for the model parameters and show that fast computation is possible even for large data sets when the observations are on a regular latitude-longitude grid. Characteristics and utility of the proposed methodology are illustrated through simulation studies and by applying it to an ocean surface wind velocity data set collected through satellite-based scatterometry remote sensing. We also compare the performance of the proposed model with a class of bivariate Mat\'ern models in terms of estimation and prediction, and demonstrate that the proposed model is superior in capturing certain physical characteristics of the wind fields.

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arXiv2016

Estimating fiber orientation distribution from diffusion MRI with spherical needlets

We present a novel method for estimation of the fiber orientation distribution (FOD) function based on diffusion-weighted Magnetic Resonance Imaging (D-MRI) data. We formulate the problem of FOD estimation as a regression problem through spherical deconvolution and a sparse representation of the FOD by a spherical needlets basis that form a multi-resolution tight frame for spherical functions. This sparse representation allows us to estimate FOD by an $l_1$-penalized regression under a non-negativity constraint. The resulting convex optimization problem is solved by an alternating direction method of multipliers (ADMM) algorithm. The proposed method leads to a reconstruction of the FODs that is accurate, has low variability and preserves sharp features. Through extensive experiments, we demonstrate the effectiveness and favorable performance of the proposed method compared with two existing methods. Particularly, we show the ability of the proposed method in successfully resolving fiber crossing at small angles and in automatically identifying isotropic diffusion. We also apply the proposed method to real 3T D-MRI data sets of healthy elderly individuals. The results show realistic descriptions of crossing fibers that are more accurate and less noisy than competing methods even with a relatively small number of gradient directions.

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arXiv2016

An adaptable generalization of Hotelling's $T^2$ test in high dimension

We propose a two-sample test for detecting the difference between mean vectors in a high-dimensional regime based on a ridge-regularized Hotelling's $T^2$. To choose the regularization parameter, a method is derived that aims at maximizing power within a class of local alternatives. We also propose a composite test that combines the optimal tests corresponding to a specific collection of local alternatives. Weak convergence of the stochastic process corresponding to the ridge-regularized Hotelling's $T^2$ is established and used to derive the cut-off values of the proposed test. Large sample properties are verified for a class of sub-Gaussian distributions. Through an extensive simulation study, the composite test is shown to compare favorably against a host of existing two-sample test procedure in a wide range of settings. The performance of the proposed test procedure is illustrated through an application to a breast cancer data set where the goal is to detect the pathways with different DNA copy number alterations across breast cancer subtypes.

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arXiv2015

Spectral analysis of linear time series in moderately high dimensions

This article is concerned with the spectral behavior of $p$-dimensional linear processes in the moderately high-dimensional case when both dimensionality $p$ and sample size $n$ tend to infinity so that $p/n\to0$. It is shown that, under an appropriate set of assumptions, the empirical spectral distributions of the renormalized and symmetrized sample autocovariance matrices converge almost surely to a nonrandom limit distribution supported on the real line. The key assumption is that the linear process is driven by a sequence of $p$-dimensional real or complex random vectors with i.i.d. entries possessing zero mean, unit variance and finite fourth moments, and that the $p\times p$ linear process coefficient matrices are Hermitian and simultaneously diagonalizable. Several relaxations of these assumptions are discussed. The results put forth in this paper can help facilitate inference on model parameters, model diagnostics and prediction of future values of the linear process.

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