Session 33: Spectral Clustering, Functional Graphical Models, and Hierarchical Interactions

Session title: Spectral Clustering, Functional Graphical Models, and Hierarchical Interactions
Organizer: Lingzhou Xue (Penn State)
Chair: Kuang-Yao Lee (Temple)
Time: June 5th, 3:15pm – 4:45pm
Location: VEC 1402

Speech 1: Spectral clustering based on learning similarity matrix 
Speaker: Hongyu Zhao (Yale)
Abstract:  Single-cell RNA-sequencing (scRNA-seq) technology can generate genome-wide expression data at the single-cell levels. One important objective in scRNA-seq analysis is to cluster cells where each cluster consists of cells belonging to the same cell type based on gene expression patterns. In this presentation, we will introduce a novel spectral clustering framework that imposes sparse structures on a target matrix. Specifically, we utilize multiple doubly stochastic similarity matrices to learn a similarity matrix, motivated by the observation that each similarity matrix can be a different informative representation of the data. We impose a sparse structure on the target matrix followed by shrinking pairwise differences of the rows in the target matrix, motivated by the fact that the target matrix should have these structures in the ideal case. We solve the proposed non-convex problem iteratively using the ADMM algorithm and show the convergence of the algorithm. We evaluate the performance of the proposed clustering method on various simulated as well as real scRNA-seq data, and show that it can identify clusters accurately and robustly. This is joint work with Seyoung Park.

Speech 2: Copula Gaussian Graphical Models for Functional Data 
Speaker: Bing Li (Penn State)
Abstract: We consider the problem of constructing statistical graphical models for functional data; that is, the observations on the vertices are random functions. This types of data are common in medical applications such as EEG and fMRI. Recently published functional graphical models rely on the assumption that the random functions are Hilbert-space-valued Gaussian random elements. We relax this assumption by introducing a copula Gaussian random elements Hilbert spaces, leading to what we call the Functional Copula Gaussian Graphical Model (FCGGM). This model removes the marginal Gaussian assumption but retains the simplicity of the Gaussian dependence structure, which is particularly attractive for large data. We develop four estimators, together with their implementation algorithms, for the FCGGM. We establish the consistency and the convergence rates of one of the estimators under different sets of sufficient conditions with varying strengths. We compare our FCGGM with the existing functional Gaussian graphical model by simulation, under both non-Gaussian and Gaussian graphical models, and apply our method to an EEG data set to construct brain networks. This is a joint work with Eftychia Solea.

Speech 3: Learning Nonconvex Hierarchical Interactions
Speaker: Lingzhou Xue (Penn State)
Abstract: In this talk, we will focus on learning nonconvex hierarchical interactions in high-dimensional statistical models. We first use the affine sparsity constraints to provide a precise characterization of both strong and weak hierarchical interactions. However, these affine sparsity constraints do not lead to a closed feasible region. To address this issue, we derive the explicit closure of the affine sparsity constraint for learning nonconvex hierarchical interactions. We prove that the global solution can be found by solving a sequence of folded concave penalized estimation and the desired strong or weak hierarchy holds with probability one. Furthermore, we study the local convergence theory for learning hierarchical interactions using the folded concave penalized estimation. Numerical studies are used to demonstrate the power of our proposed methods.