Innovated interaction screening for high-dimensional nonlinear classification

TL;DR

This paper introduces IIS-SQDA, a two-step method for efficient interaction screening and nonlinear classification in high-dimensional data, demonstrating theoretical guarantees and superior empirical performance.

Contribution

The paper proposes a novel two-step approach combining innovated interaction screening with sparse quadratic discriminant analysis for high-dimensional nonlinear classification.

Findings

The method enjoys sure screening property in high-dimensional settings.

Classification error is bounded by the oracle error plus a small term.

Outperforms existing methods in simulations and real data analysis.

Abstract

This paper is concerned with the problems of interaction screening and nonlinear classification in a high-dimensional setting. We propose a two-step procedure, IIS-SQDA, where in the first step an innovated interaction screening (IIS) approach based on transforming the original $p$-dimensional feature vector is proposed, and in the second step a sparse quadratic discriminant analysis (SQDA) is proposed for further selecting important interactions and main effects and simultaneously conducting classification. Our IIS approach screens important interactions by examining only $p$ features instead of all two-way interactions of order $O(p^2)$. Our theory shows that the proposed method enjoys sure screening property in interaction selection in the high-dimensional setting of $p$ growing exponentially with the sample size. In the selection and classification step, we establish a sparse inequality on the estimated coefficient vector for QDA and prove that the classification error of our procedure can be upper-bounded by the oracle classification error plus some smaller order term. Extensive simulation studies and real data analysis show that our proposal compares favorably with existing methods in interaction selection and high-dimensional classification.