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1. T-LoHo: A Bayesian Regularization Model for Structured Sparsity and Smoothness on Graphs

   Lee, Changwoo; Zhao Tang Luo; and Huiyan Sang (December 2021, Advances in neural information processing systems)

   Graphs have been commonly used to represent complex data structures. In models dealing with graph-structured data, multivariate parameters may not only exhibit sparse patterns but have structured sparsity and smoothness in the sense that both zero and non-zero parameters tend to cluster together. We propose a new prior for high-dimensional parameters with graphical relations, referred to as the Tree-based Low-rank Horseshoe (T-LoHo) model, that generalizes the popular univariate Bayesian horseshoe shrinkage prior to the multivariate setting to detect structured sparsity and smoothness simultaneously. The T-LoHo prior can be embedded in many high-dimensional hierarchical models. To illustrate its utility, we apply it to regularize a Bayesian high-dimensional regression problem where the regression coefficients are linked by a graph, so that the resulting clusters have flexible shapes and satisfy the cluster contiguity constraint with respect to the graph. We design an efficient Markov chain Monte Carlo algorithm that delivers full Bayesian inference with uncertainty measures for model parameters such as the number of clusters. We offer theoretical investigations of the clustering effects and posterior concentration results. Finally, we illustrate the performance of the model with simulation studies and a real data application for anomaly detection on a road network. The results indicate substantial improvements over other competing methods such as the sparse fused lasso. 

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2. Multi-Attribute Graph Estimation With Sparse-Group Non-Convex Penalties

   https://doi.org/10.1109/ACCESS.2025.3566959  

   Tugnait, Jitendra K (January 2025, IEEE Access)

   We consider the problem of inferring the conditional independence graph (CIG) of high-dimensional Gaussian vectors from multi-attribute data. Most existing methods for graph estimation are based on single-attribute models where one associates a scalar random variable with each node. In multi-attribute graphical models, each node represents a random vector. In this paper we provide a unified theoretical analysis of multi-attribute graph learning using a penalized log-likelihood objective function. We consider both convex (sparse-group lasso) and sparse-group non-convex (log-sum and smoothly clipped absolute deviation (SCAD) penalties) penalty/regularization functions. An alternating direction method of multipliers (ADMM) approach coupled with local linear approximation to non-convex penalties is presented for optimization of the objective function. For non-convex penalties, theoretical analysis establishing local consistency in support recovery, local convexity and precision matrix estimation in high-dimensional settings is provided under two sets of sufficient conditions: with and without some irrepresentability conditions. We illustrate our approaches using both synthetic and real-data numerical examples. In the synthetic data examples the sparse-group log-sum penalized objective function significantly outperformed the lasso penalized as well as SCAD penalized objective functions with F1 -score and Hamming distance as performance metrics. 

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3. Manifold filter-combine networks

   https://doi.org/10.1007/s43670-025-00115-2  

   Johnson, David R; Chew, Joyce A; Viswanath, Siddharth; Brouwer, Edward De; Needell, Deanna; Krishnaswamy, Smita; Perlmutter, Michael (December 2025, Sampling Theory, Signal Processing, and Data Analysis)

   Abstract In order to better understand manifold neural networks (MNNs), we introduce Manifold Filter-Combine Networks (MFCNs). Our filter-combine framework parallels the popular aggregate-combine paradigm for graph neural networks (GNNs) and naturally suggests many interesting families of MNNs which can be interpreted as manifold analogues of various popular GNNs. We propose a method for implementing MFCNs on high-dimensional point clouds that relies on approximating an underlying manifold by a sparse graph. We then prove that our method is consistent in the sense that it converges to a continuum limit as the number of data points tends to infinity, and we numerically demonstrate its effectiveness on real-world and synthetic data sets. 

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4. A neural kernel method for capturing multiscale high-dimensional micromorphic plasticity of materials with internal structures

   https://doi.org/10.1016/j.cma.2023.116317  

   Xiong, Zeyu; Xiao, Mian; Vlassis, Nikolaos; Sun, WaiChing (November 2023, Computer Methods in Applied Mechanics and Engineering)

   De Lorenzis, L; Papadrakakis, M; Zohdi T.I. (Ed.)

   This paper introduces a neural kernel method to generate machine learning plasticity models for micropolar and micromorphic materials that lack material symmetry and have internal structures. Since these complex materials often require higher-dimensional parametric space to be precisely characterized, we introduce a representation learning step where we first learn a feature vector space isomorphic to a finite-dimensional subspace of the original parametric function space from the augmented labeled data expanded from the narrow band of the yield data. This approach simplifies the data augmentation step and enables us to constitute the high-dimensional yield surface in a feature space spanned by the feature kernels. In the numerical examples, we first verified the implementations with data generated from known models, then tested the capacity of the models to discover feature spaces from meso-scale simulation data generated from representative elementary volume (RVE) of heterogeneous materials with internal structures. The neural kernel plasticity model and other alternative machine learning approaches are compared in a computational homogenization problem for layered geomaterials. The results indicate that the neural kernel feature space may lead to more robust forward predictions against sparse and high-dimensional data. 

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5. An Iterative Penalized Least Squares Approach to Sparse Canonical Correlation Analysis

   https://doi.org/10.1111/biom.13043  

   Mai, Qing; Zhang, Xin (February 2019, Biometrics)

   Abstract It is increasingly interesting to model the relationship between two sets of high-dimensional measurements with potentially high correlations. Canonical correlation analysis (CCA) is a classical tool that explores the dependency of two multivariate random variables and extracts canonical pairs of highly correlated linear combinations. Driven by applications in genomics, text mining, and imaging research, among others, many recent studies generalize CCA to high-dimensional settings. However, most of them either rely on strong assumptions on covariance matrices, or do not produce nested solutions. We propose a new sparse CCA (SCCA) method that recasts high-dimensional CCA as an iterative penalized least squares problem. Thanks to the new iterative penalized least squares formulation, our method directly estimates the sparse CCA directions with efficient algorithms. Therefore, in contrast to some existing methods, the new SCCA does not impose any sparsity assumptions on the covariance matrices. The proposed SCCA is also very flexible in the sense that it can be easily combined with properly chosen penalty functions to perform structured variable selection and incorporate prior information. Moreover, our proposal of SCCA produces nested solutions and thus provides great convenient in practice. Theoretical results show that SCCA can consistently estimate the true canonical pairs with an overwhelming probability in ultra-high dimensions. Numerical results also demonstrate the competitive performance of SCCA. 

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