GENERALIZED MATRIX DECOMPOSITION REGRESSION: ESTIMATION AND INFERENCE FOR TWO-WAY STRUCTURED DATA.

Area of Science:

• Statistics
• Machine Learning
• Data Science

Background:

• High-dimensional regression is crucial for complex datasets in ecology, microbiology, and neuroscience.
• Existing methods often struggle with two-way structured data, limiting their application.
• Leveraging auxiliary information on data structure can enhance model performance.

Purpose of the Study:

• To develop novel methods for high-dimensional regression with two-way structured data.
• To propose an efficient estimation technique (GMDR) and a flexible inference framework (GMDI).
• To provide accurate prediction and reliable inference for individual regression coefficients.

Main Methods:

• Generalized Matrix Decomposition Regression (GMDR): Extends Principal Component Regression (PCR) by selecting predictive components for two-way structured data.
• Generalized Matrix Decomposition Inference (GMDI): A general framework for high-dimensional inference, accommodating GMDR and other estimators.
• GMDI allows for dependent and heteroscedastic observations and constrains coefficient representations based on column structure.

Main Results:

• GMDR demonstrates improved prediction accuracy compared to traditional PCR.
• GMDI provides theoretical guarantees on type-I error rate and power for statistical inference.
• Simulations and a human microbiome data application confirm the effectiveness of GMDR and GMDI.

Conclusions:

• GMDR and GMDI offer powerful and flexible tools for analyzing high-dimensional two-way structured data.
• These methods advance statistical modeling in fields with complex data structures.
• The proposed frameworks enable more accurate predictions and robust inference in emerging scientific applications.