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Adversarial machine learning has attracted a great amount of attention in recent years. In a poisoning attack, the adversary can inject a small number of specially crafted samples into the training data which make the decision boundary severely deviate and cause unexpected misclassification. Due to the great importance and popular use of support vector machines (SVM), we consider defending SVM against poisoning attacks in this paper. We study two commonly used strategies for defending: designing robust SVM algorithms and data sanitization. Though several robust SVM algorithms have been proposed before, most of them either are in lack of adversarial-resilience, or rely on strong assumptions about the data distribution or the attacker's behavior. Moreover, the research on their complexities is still quite limited. We are the first, to the best of our knowledge, to prove that even the simplest hard-margin one-class SVM with outliers problem is NP-complete, and has no fully PTAS unless P$=$NP (that means it is hard to achieve an even approximate algorithm). For the data sanitization defense, we link it to the intrinsic dimensionality of data; in particular, we provide a sampling theorem in doubling metrics for explaining the effectiveness of DBSCAN (as a density-based outlier removal method) for defending against poisoning attacks. In our empirical experiments, we compare several defenses including the DBSCAN and robust SVM methods, and investigate the influences from the intrinsic dimensionality and data density to their performances.