Sparse Principal Component Analysis with Non-Oblivious Adversarial Perturbations

TL;DR

This paper develops a robust sparse PCA method resilient to non-oblivious adversarial perturbations, providing theoretical guarantees and practical algorithms for high-dimensional data analysis.

Contribution

It introduces a new formulation for sparse PCA under non-oblivious adversarial perturbations and derives MIP-based bounds with provable guarantees.

Findings

MIP reformulations upper bound the robust sparse PCA problem.

The method guarantees vector recovery under the spiked Wishart model.

Numerical results validate the theoretical robustness and accuracy.

Abstract

Sparse Principal Component Analysis (sparse PCA) is a fundamental dimension-reduction tool that enhances interpretability in various high-dimensional settings. An important variant of sparse PCA studies the scenario when samples are adversarially perturbed. Notably, most existing statistical studies on this variant focus on recovering the ground truth and verifying the robustness of classical algorithms when the given samples are corrupted under oblivious adversarial perturbations. In contrast, this paper aims to find a robust sparse principal component that maximizes the variance of the given samples corrupted by non-oblivious adversarial perturbations, say sparse PCA with Non-Oblivious Adversarial Perturbations (sparse PCA-NOAP). Specifically, we introduce a general formulation for the proposed sparse PCA-NOAP. We then derive Mixed-Integer Programming (MIP) reformulations to upper bound it with provable worst-case guarantees when adversarial perturbations are controlled by two typical norms, i.e., $\ell_{2 \rightarrow \infty}$-norm (sample-wise $\ell_2$-norm perturbation) and $\ell_{1 \rightarrow 2}$-norm (feature-wise $\ell_2$-norm perturbation). Moreover, when samples are drawn from the spiked Wishart model, we show that the proposed MIP reformulations ensure vector recovery properties under a more general parameter region compared with existing results. Numerical simulations are also provided to validate the theoretical findings and demonstrate the accuracy of the proposed formulations.