# Manifold Assumption and Defenses Against Adversarial Perturbations

In the adversarial perturbation problem of neural networks, an adversary starts with a neural network model F and a point x that F classifies correctly, and identifies another point x', which is nearby x, that F classifies incorrectly. In this paper we consider a defense method that is based on the semantics of F. Our starting point is the common manifold assumption, which states that natural data points lie on separate low dimensional manifolds for different classes. We then make a further postulate which states that (a good model) F is confident on natural points on the manifolds, but has low confidence on points outside of the manifolds, where a natural measure of "confident behavior" is F( x)_∞ (i.e. how confident F is about its prediction). Under this postulate, an adversarial example becomes a point that is outside of the low dimensional manifolds which F has learned, but is still close to at least one manifold under some distance metric. Therefore, defending against adversarial perturbations becomes embedding an adversarial point back to the nearest manifold where natural points are drawn from. We propose algorithms to formalize this intuition and perform a preliminary evaluation. Noting that the effectiveness of our method depends on both how well F satisfies the postulate and how effective we can conduct the embedding, we use a model trained recently by Madry et al., as the base model, and use gradient based optimization, such as the Carlini-Wagner attack (but now they are used for defense), as the embedding procedure. Our preliminary results are encouraging: The base model wrapped with the embedding procedure achieves almost perfect success rate in defending against attacks that the base model fails on, while retaining the good generalization behavior of the base model.

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