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The plane-wave pseudopotential (PW-PP) formalism is widely used for the first-principles electronic structure calculation of extended periodic systems. The PW-PP approach has also been adapted for real-time time-dependent density functional theory (RT-TDDFT) to investigate time-dependent electronic dynamical phenomena. In this work, we detail recent advances in the PW-PP formalism for RT-TDDFT, particularly how maximally localized Wannier functions (MLWFs) are used to accelerate simulations using the exact exchange. We also discuss several related developments, including an anti-Hermitian correction for the time-dependent MLWFs (TD-MLWFs) when a time-dependent electric field is applied, the refinement procedure for TD-MLWFs, comparison of the velocity and length gauge approaches for applying an electric field, and elimination of long-range electrostatic interaction, as well as usage of a complex absorbing potential for modeling isolated systems when using the PW-PP formalism. 

The adversarial risk of a machine learning model has been widely studied. Most previous studies assume that the data lie in the whole ambient space. We propose to take a new angle and take the manifold assumption into consideration. Assuming data lie in a manifold, we investigate two new types of adversarial risk, the normal adversarial risk due to perturbation along normal direction and the in-manifold adversarial risk due to perturbation within the manifold. We prove that the classic adversarial risk can be bounded from both sides using the normal and in-manifold adversarial risks. We also show a surprisingly pessimistic case that the standard adversarial risk can be non-zero even when both normal and in-manifold adversarial risks are zero. We finalize the study with empirical studies supporting our theoretical results. Our results suggest the possibility of improving the robustness of a classifier without sacrificing model accuracy, by only focusing on the normal adversarial risk.