Robustness is a key requirement for widespread deployment of machine learning algorithms, and has received much attention in both statistics and computer science. We study a natural model of robustness for high-dimensional statistical estimation problems that we call the adversarial perturbation model. An adversary can perturb every sample arbitrarily up to a specified magnitude $\delta$ measured in some $\ell_q$ norm, say $\ell_\infty$. Our model is motivated by emerging paradigms such as low precision machine learning and adversarial training. We study the classical problem of estimating the top-$r$ principal subspace of the Gaussian covariance matrix in high dimensions, under the adversarial perturbation model. We design a computationally efficient algorithm that given corrupted data, recovers an estimate of the top-$r$ principal subspace with error that depends on a robustness parameter $\kappa$ that we identify. This parameter corresponds to the $q \to 2$ operator norm of the projector onto the principal subspace, and generalizes well-studied analytic notions of sparsity. Additionally, in the absence of corruptions, our algorithmic guarantees recover existing bounds for problems such as sparse PCA and its higher rank analogs. We also prove that the above dependence on the parameter $\kappa$ is almost optimal asymptotically, not just in a minimax sense, but remarkably for every instance of the problem. This instance-optimal guarantee shows that the $q \to 2$ operator norm of the subspace essentially characterizes the estimation error under adversarial perturbations.

Pose variation is one of the challenging factors for face recognition. In this paper, we propose a novel cross-pose face recognition method named as Regularized Latent Least Square Regression (RLLSR). The basic assumption is that the images captured under different poses of one person can be viewed as pose-specific transforms of a single ideal object. We treat the observed images as regressor, the ideal object as response, and then formulate this assumption in the least square regression framework, so as to learn the multiple pose-specific transforms. Specifically, we incorporate some prior knowledge as two regularization terms into the least square approach: 1) the smoothness regularization, as the transforms for nearby poses should not differ too much; 2) the local consistency constraint, as the distribution of the latent ideal objects should preserve the geometric structure of the observed image space. We develop an alternating algorithm to simultaneously solve for the ideal objects of the training individuals and a set of pose-specific transforms. The experimental results on the Multi-PIE dataset demonstrate the effectiveness of the proposed method and superiority over the previous methods.