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Adversarial examples are crafted by adding indistinguishable perturbations tonormal examples in order to fool a well-trained deep learning model tomisclassify. In the context of computer vision, this notion ofindistinguishability is typically bounded by $L_{\infty}$ or other norms.However, these norms are not appropriate for measuring indistinguishiabilityfor time series data. In this work, we propose adversarial examples in theWasserstein space for time series data for the first time and utilizeWasserstein distance to bound the perturbation between normal examples andadversarial examples. We introduce Wasserstein projected gradient descent(WPGD), an adversarial attack method for perturbing univariant time seriesdata. We leverage the closed-form solution of Wasserstein distance in the 1Dspace to calculate the projection step of WPGD efficiently with the gradientdescent method. We further propose a two-step projection so that the search ofadversarial examples in the Wasserstein space is guided and constrained byEuclidean norms to yield more effective and imperceptible perturbations. Weempirically evaluate the proposed attack on several time series datasets in thehealthcare domain. Extensive results demonstrate that the Wasserstein attack ispowerful and can successfully attack most of the target classifiers with a highattack success rate. To better study the nature of Wasserstein adversarialexample, we evaluate a strong defense mechanism named Wasserstein smoothing forpotential certified robustness defense. Although the defense can achieve someaccuracy gain, it still has limitations in many cases and leaves space fordeveloping a stronger certified robustness method to Wasserstein adversarialexamples on univariant time series data.

Wasserstein Adversarial Examples on Univariant Time Series Data