Tensor Graphical Model: Non-Convex Optimization and Statistical Inference.

Area of Science:

• Statistics
• Machine Learning
• Data Science

Background:

• High-dimensional tensor-valued data present unique challenges for modeling dependency structures.
• Existing graphical models struggle with the complexity and scale of tensor data.
• The Kronecker product structure in tensor normal distributions offers a tractable approach.

Purpose of the Study:

• To develop methods for estimating and inferring graphical models for high-dimensional tensor-valued data.
• To address the non-convex optimization problem in penalized maximum likelihood estimation.
• To enable robust statistical inference and hypothesis testing on the support of sparse precision matrices.

Main Methods:

• Utilizing a tensor normal distribution with a Kronecker product covariance structure.
• Employing an alternating minimization algorithm to overcome non-convexity in estimation.
• Developing a de-biased statistical inference procedure for hypothesis testing.
• Implementing false discovery rate (FDR) control for multiple hypothesis testing.

Main Results:

• An alternating minimization algorithm achieves an optimal statistical rate of convergence for estimator.
• The proposed de-biased inference procedure demonstrates asymptotic normality.
• The FDR control procedure is consistent, ensuring reliable hypothesis testing.
• Real-world applications yield significant scientific and business insights.

Conclusions:

• The proposed methods provide an effective framework for graphical model estimation and inference in high-dimensional tensor data.
• The Tlasso R package facilitates the application of these advanced statistical techniques.
• The approach offers valuable findings in neuroimaging and advertising click analysis.