Non-convex Statistical Optimization for Sparse Tensor Graphical Model.

Area of Science:

• Statistics
• Machine Learning
• Data Science

Background:

• High-dimensional tensor-valued data presents unique challenges for modeling dependency structures.
• Sparse graphical models are crucial for understanding complex relationships in such data.
• Existing methods often struggle with the non-convexity inherent in estimating tensor graphical models.

Purpose of the Study:

• To develop an efficient and statistically sound method for estimating sparse graphical models for tensor-valued data.
• To address the challenges posed by non-convex objective functions in penalized maximum likelihood estimation.
• To achieve optimal statistical rates of convergence and consistent graph recovery.

Main Methods:

• Utilizing a tensor normal distribution with a Kronecker product covariance structure.
• Employing an alternating minimization algorithm for penalized maximum likelihood estimation.
• Analyzing the theoretical convergence properties of the proposed estimation algorithm.

Main Results:

• The alternating minimization algorithm achieves an estimator with the optimal statistical rate of convergence.
• The method guarantees consistent recovery of the underlying sparse graphical structure.
• Estimation consistency is demonstrated with as few as one tensor sample, a significant improvement over prior work.

Conclusions:

• The proposed method offers a robust solution for sparse graphical model estimation in high-dimensional tensor data.
• The algorithm's ability to achieve consistency with minimal data is a key theoretical advancement.
• Numerical studies validate the theoretical findings, demonstrating practical applicability.