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1. Integrative Multi-View Regression: Bridging Group-Sparse and Low-Rank Models

   https://doi.org/10.1111/biom.13006  

   Li, Gen ; Liu, Xiaokang ; Chen, Kun ( November 2018 , Biometrics)

   Multi-view data have been routinely collected in various fields of science and engineering. A general problem is to study the predictive association between multivariate responses and multi-view predictor sets, all of which can be of high dimensionality. It is likely that only a few views are relevant to prediction, and the predictors within each relevant view contribute to the prediction collectively rather than sparsely. We cast this new problem under the familiar multivariate regression framework and propose an integrative reduced-rank regression (iRRR), where each view has its own low-rank coefficient matrix. As such, latent features are extracted from each view in a supervised fashion. For model estimation, we develop a convex composite nuclear norm penalization approach, which admits an efficient algorithm via alternating direction method of multipliers. Extensions to non-Gaussian and incomplete data are discussed. Theoretically, we derive non-asymptotic oracle bounds of iRRR under a restricted eigenvalue condition. Our results recover oracle bounds of several special cases of iRRR including Lasso, group Lasso, and nuclear norm penalized regression. Therefore, iRRR seamlessly bridges group-sparse and low-rank methods and can achieve substantially faster convergence rate under realistic settings of multi-view learning. Simulation studies and an application in the Longitudinal Studies of Aging further showcase the efficacy of the proposed methods.

    

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2. Multivariate log‐contrast regression with sub‐compositional predictors: Testing the association between preterm infants' gut microbiome and neurobehavioral outcomes

   https://doi.org/10.1002/sim.9273  

   Liu, Xiaokang ; Cong, Xiaomei ; Li, Gen ; Maas, Kendra ; Chen, Kun ( December 2021 , Statistics in Medicine)

   To link a clinical outcome with compositional predictors in microbiome analysis, the linear log‐contrast model is a popular choice, and the inference procedure for assessing the significance of each covariate is also available. However, with the existence of multiple potentially interrelated outcomes and the information of the taxonomic hierarchy of bacteria, a multivariate analysis method that considers the group structure of compositional covariates and an accompanying group inference method are still lacking. Motivated by a study for identifying the microbes in the gut microbiome of preterm infants that impact their later neurobehavioral outcomes, we formulate a constrained integrative multi‐view regression. The neurobehavioral scores form multivariate responses, the log‐transformed sub‐compositional microbiome data form multi‐view feature matrices, and a set of linear constraints on their corresponding sub‐coefficient matrices ensures the sub‐compositional nature. We assume all the sub‐coefficient matrices are possible of low‐rank to enable joint selection and inference of sub‐compositions/views. We propose a scaled composite nuclear norm penalization approach for model estimation and develop a hypothesis testing procedure through de‐biasing to assess the significance of different views. Simulation studies confirm the effectiveness of the proposed procedure. We apply the method to the preterm infant study, and the identified microbes are mostly consistent with existing studies and biological understandings.

    

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3. Asymptotic properties of adaptive group Lasso for sparse reduced rank regression

   https://doi.org/10.1002/sta4.123  

   He, Kejun ; Huang, Jianhua Z. ( October 2016 , Stat)

   This paper studies the asymptotic properties of the penalized least squares estimator using an adaptive group Lasso penalty for the reduced rank regression. The group Lasso penalty is defined in the way that the regression coefficients corresponding to each predictor are treated as one group. It is shown that under certain regularity conditions, the estimator can achieve the minimax optimal rate of convergence. Moreover, the variable selection consistency can also be achieved, that is, the relevant predictors can be identified with probability approaching one. In the asymptotic theory, the number of response variables, the number of predictors and the rank number are allowed to grow to infinity with the sample size. Copyright © 2016 John Wiley & Sons, Ltd.

    

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4. Multivariate Sparse Group Lasso for the Multivariate Multiple Linear Regression with an Arbitrary Group Structure

   https://doi.org/10.1111/biom.12292  

   Li, Yanming ; Nan, Bin ; Zhu, Ji ( March 2015 , Biometrics)

   We propose a multivariate sparse group lasso variable selection and estimation method for data with high-dimensional predictors as well as high-dimensional response variables. The method is carried out through a penalized multivariate multiple linear regression model with an arbitrary group structure for the regression coefficient matrix. It suits many biology studies well in detecting associations between multiple traits and multiple predictors, with each trait and each predictor embedded in some biological functional groups such as genes, pathways or brain regions. The method is able to effectively remove unimportant groups as well as unimportant individual coefficients within important groups, particularly for large p small n problems, and is flexible in handling various complex group structures such as overlapping or nested or multilevel hierarchical structures. The method is evaluated through extensive simulations with comparisons to the conventional lasso and group lasso methods, and is applied to an eQTL association study.

    

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5. Hierarchical Nuclear Norm Penalization for Multi-View Data Integration

   https://doi.org/10.1111/biom.13893  

   Yi, Sangyoon ; Wong, Raymond Ka Wai ; Gaynanova, Irina ( June 2023 , Biometrics)

   The prevalence of data collected on the same set of samples from multiple sources (i.e., multi-view data) has prompted significant development of data integration methods based on low-rank matrix factorizations. These methods decompose signal matrices from each view into the sum of shared and individual structures, which are further used for dimension reduction, exploratory analyses, and quantifying associations across views. However, existing methods have limitations in modeling partially-shared structures due to either too restrictive models, or restrictive identifiability conditions. To address these challenges, we propose a new formulation for signal structures that include partially-shared signals based on grouping the views into so-called hierarchical levels with identifiable guarantees under suitable conditions. The proposed hierarchy leads us to introduce a new penalty, hierarchical nuclear norm (HNN), for signal estimation. In contrast to existing methods, HNN penalization avoids scores and loadings factorization of the signals and leads to a convex optimization problem, which we solve using a dual forward–backward algorithm. We propose a simple refitting procedure to adjust the penalization bias and develop an adapted version of bi-cross-validation for selecting tuning parameters. Extensive simulation studies and analysis of the genotype-tissue expression data demonstrate the advantages of our method over existing alternatives.

    

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