Transfer Learning in Large-Scale Gaussian Graphical Models with False Discovery Rate Control

Transfer learning for high-dimensional Gaussian graphical models (GGMs) is studied. The target GGM is estimated by incorporating the data from similar and related auxiliary studies, where the similarity between the target graph and each auxiliary graph is characterized by the sparsity of a divergence matrix. An estimation algorithm, Trans-CLIME, is proposed and shown to attain a faster convergence rate than the minimax rate in the single-task setting. Furthermore, we introduce a universal debiasing method that can be coupled with a range of initial graph estimators and can be analytically computed in one step. A debiased Trans-CLIME estimator is then constructed and is shown to be element-wise asymptotically normal. This fact is used to construct a multiple testing procedure for edge detection with false discovery rate control. The proposed estimation and multiple testing procedures demonstrate superior numerical performance in simulations and are applied to infer the gene networks in a target brain tissue by leveraging the gene expressions from multiple other brain tissues. A significant decrease in prediction errors and a significant increase in power for link detection are observed. Supplementary materials for this article are available online.</p

GAP: A General Framework for Information Pooling in Two-Sample Sparse Inference

This article develops a general framework for exploiting the sparsity information in two-sample multiple testing problems. We propose to first construct a covariate sequence, in addition to the usual primary test statistics, to capture the sparsity structure, and then incorporate the auxiliary covariates in inference via a three-step algorithm consisting of grouping, adjusting and pooling (GAP). The GAP procedure provides a simple and effective framework for information pooling. An important advantage of GAP is its capability of handling various dependence structures such as those arise from high-dimensional linear regression, differential correlation analysis, and differential network analysis. We establish general conditions under which GAP is asymptotically valid for false discovery rate control, and show that these conditions are fulfilled in a range of settings, including testing multivariate normal means, high-dimensional linear regression, differential covariance or correlation matrices, and Gaussian graphical models. Numerical results demonstrate that existing methods can be significantly improved by the proposed framework. The GAP procedure is illustrated using a breast cancer study for identifying gene–gene interactions.</p

Global and Simultaneous Hypothesis Testing for High-Dimensional Logistic Regression Models

High-dimensional logistic regression is widely used in analyzing data with binary outcomes. In this article, global testing and large-scale multiple testing for the regression coefficients are considered in both single- and two-regression settings. A test statistic for testing the global null hypothesis is constructed using a generalized low-dimensional projection for bias correction and its asymptotic null distribution is derived. A lower bound for the global testing is established, which shows that the proposed test is asymptotically minimax optimal over some sparsity range. For testing the individual coefficients simultaneously, multiple testing procedures are proposed and shown to control the false discovery rate and falsely discovered variables asymptotically. Simulation studies are carried out to examine the numerical performance of the proposed tests and their superiority over existing methods. The testing procedures are also illustrated by analyzing a dataset of a metabolomics study that investigates the association between fecal metabolites and pediatric Crohn’s disease and the effects of treatment on such associations. Supplementary materials for this article are available online.</p

Statistical Inference for High-Dimensional Generalized Linear Models With Binary Outcomes

This article develops a unified statistical inference framework for high-dimensional binary generalized linear models (GLMs) with general link functions. Both unknown and known design distribution settings are considered. A two-step weighted bias-correction method is proposed for constructing confidence intervals (CIs) and simultaneous hypothesis tests for individual components of the regression vector. Minimax lower bound for the expected length is established and the proposed CIs are shown to be rate-optimal up to a logarithmic factor. The numerical performance of the proposed procedure is demonstrated through simulation studies and an analysis of a single cell RNA-seq dataset, which yields interesting biological insights that integrate well into the current literature on the cellular immune response mechanisms as characterized by single-cell transcriptomics. The theoretical analysis provides important insights on the adaptivity of optimal CIs with respect to the sparsity of the regression vector. New lower bound techniques are introduced and they can be of independent interest to solve other inference problems in high-dimensional binary GLMs.</p

Sparse Topic Modeling: Computational Efficiency, Near-Optimal Algorithms, and Statistical Inference

Sparse topic modeling under the probabilistic latent semantic indexing (pLSI) model is studied. Novel and computationally fast algorithms for estimation and inference of both the word-topic matrix and the topic-document matrix are proposed and their theoretical properties are investigated. Both minimax upper and lower bounds are established and the results show that the proposed algorithms are rate-optimal, up to a logarithmic factor. Moreover, a refitting algorithm is proposed to establish asymptotic normality and construct valid confidence intervals for the individual entries of the word-topic and topic-document matrices. Simulation studies are carried out to investigate the numerical performance of the proposed algorithms. The results show that the proposed algorithms perform well numerically and are more accurate in a range of simulation settings comparing to the existing literature. In addition, the methods are illustrated through an analysis of the COVID-19 Open Research Dataset (CORD-19).</p

Optimal Permutation Recovery in Permuted Monotone Matrix Model

Motivated by recent research on quantifying bacterial growth dynamics based on genome assemblies, we consider a permuted monotone matrix modelY=ΘΠ+Z, where the rows represent different samples, the columns represent contigs in genome assemblies and the elements represent log-read counts after preprocessing steps and Guanine-Cytosine (GC) adjustment. In this model, Θ is an unknown mean matrix with monotone entries for each row, Π is a permutation matrix that permutes the columns of Θ, and Z is a noise matrix. This article studies the problem of estimation/recovery of Π given the observed noisy matrix Y. We propose an estimator based on the best linear projection, which is shown to be minimax rate-optimal for both exact recovery, as measured by the 0-1 loss, and partial recovery, as quantified by the normalized Kendall’s tau distance. Simulation studies demonstrate the superior empirical performance of the proposed estimator over alternative methods. We demonstrate the methods using a synthetic metagenomics dataset of 45 closely related bacterial species and a real metagenomic dataset to compare the bacterial growth dynamics between the responders and the nonresponders of the IBD patients after 8 weeks of treatment. Supplementary materials for this article are available online.</p

Optimal Estimation of Wasserstein Distance on a Tree With an Application to Microbiome Studies

The weighted UniFrac distance, a plug-in estimator of the Wasserstein distance of read counts on a tree, has been widely used to measure the microbial community difference in microbiome studies. Our investigation however shows that such a plug-in estimator, although intuitive and commonly used in practice, suffers from potential bias. Motivated by this finding, we study the problem of optimal estimation of the Wasserstein distance between two distributions on a tree from the sampled data in the high-dimensional setting. The minimax rate of convergence is established. To overcome the bias problem, we introduce a new estimator, referred to as the moment-screening estimator on a tree (MET), by using implicit best polynomial approximation that incorporates the tree structure. The new estimator is computationally efficient and is shown to be minimax rate-optimal. Numerical studies using both simulated and real biological datasets demonstrate the practical merits of MET, including reduced biases and statistically more significant differences in microbiome between the inactive Crohn’s disease patients and the normal controls. Supplementary materials for this article are available online.</p

Estimation and Inference for High-Dimensional Generalized Linear Models with Knowledge Transfer

Transfer learning provides a powerful tool for incorporating data from related studies into a target study of interest. In epidemiology and medical studies, the classification of a target disease could borrow information across other related diseases and populations. In this work, we consider transfer learning for high-dimensional generalized linear models (GLMs). A novel algorithm, TransHDGLM, that integrates data from the target study and the source studies is proposed. Minimax rate of convergence for estimation is established and the proposed estimator is shown to be rate-optimal. Statistical inference for the target regression coefficients is also studied. Asymptotic normality for a debiased estimator is established, which can be used for constructing coordinate-wise confidence intervals of the regression coefficients. Numerical studies show significant improvement in estimation and inference accuracy over GLMs that only use the target data. The proposed methods are applied to a real data study concerning the classification of colorectal cancer using gut microbiomes, and are shown to enhance the classification accuracy in comparison to methods that only use the target data.</p