Interaction Screening for Ultra-High Dimensional Data.

Area of Science:

• Statistics
• Machine Learning
• Data Mining

Background:

• Ultra-high dimensional data analysis presents significant challenges in identifying interaction effects.
• The computational cost of analyzing all possible interactions, especially order-2 terms, is often prohibitive for standard computational resources.

Purpose of the Study:

• To develop computationally feasible and theoretically sound methods for selecting interaction effects in ultra-high dimensional settings.
• To address the difficulties in achieving interaction selection consistency and the complex covariance structures associated with interaction terms.

Main Methods:

• Introduction of iFOR (interaction Forward Selection), a greedy, forward-selection based procedure for identifying interaction effects.
• Study of two specific algorithms, iFORT and iFORM, designed for efficiency and minimal computational requirements.
• The methods avoid storing the full augmented matrix, utilizing only OLS-type calculations for linear computational complexity.

Main Results:

• The iFOR procedures are computationally efficient, requiring minimal memory and CPU.
• Demonstrated linear computational complexity with respect to the number of predictors (p) for sparse models, making them feasible for p >> n.
• Theoretical proof of the sure screening property for ultra-high dimensional settings.

Conclusions:

• The iFOR methods provide a practical and efficient solution for identifying interaction effects in ultra-high dimensional data.
• The proposed algorithms maintain the hierarchical model structure while offering computational feasibility and theoretical guarantees.
• Numerical examples confirm the effectiveness of iFOR in finite sample performance.