Toward Robust Discriminative Projections Learning Against Adversarial Patch Attacks

As one of the most popular supervised dimensionality reduction methods, linear discriminant analysis (LDA) has been widely studied in machine learning community and applied to many scientific applications. Traditional LDA minimizes the ratio of squared
$\ell_2$
norms, which is vulnerable to the adversarial examples. In recent studies, many
$\ell_1$
-norm-based robust dimensionality reduction methods are proposed to improve the robustness of model. However, due to the difficulty of
$\ell_1$
-norm ratio optimization and weakness on defending a large number of adversarial examples, so far, scarce works have been proposed to utilize sparsity-inducing norms for LDA objective. In this article, we propose a novel robust discriminative projections learning (rDPL) method based on the
$\ell_{1,2}$
-norm trace-ratio minimization optimization algorithm. Minimizing the
$\ell_{1,2}$
-norm ratio problem directly is a much more challenging problem than the traditional methods, and there is no existing optimization algorithm to solve such nonsmooth terms ratio problem. We derive a new efficient algorithm to solve this challenging problem and provide a theoretical analysis on the convergence of our algorithm. The proposed algorithm is easy to implement and converges fast in practice. Extensive experiments on both synthetic data and several real benchmark datasets show the effectiveness of the proposed method on defending the adversarial patch attack by comparison with many state-of-the-art robust dimensionality reduction methods.