--Adversarial examples have attracted significant attention over the years, yet understanding their frequency-based characteristics remains insufficient. In this paper, we investigate the intriguing properties of adversarial examples in the frequency domain for the image classification task, with the following key findings. These results suggest that different network architectures have different frequency preferences and that differences in frequency components between adversarial and natural examples may directly influence model robustness. Based on our findings, we further conclude with three useful proposals that serve as a valuable reference to the AI model security community. Despite the fact that deep neural networks (DNNs) achieve remarkable performance in many fields [1]-[3], their counterintuitive vulnerability attracts increasing attention, both for safety-critical applications [4], [5] and the black-box mechanism of DNNs [6], [7].

This work starts with the intention of using mathematics to understand the intriguing vulnerability observed by ~\citet{szegedy2013} within artificial neural networks. Along the way, we will develop some novel tools with applications far outside of just the adversarial domain. We will do this while developing a rigorous mathematical framework to examine this problem. Our goal is to build out theory which can support increasingly sophisticated conjecture about adversarial attacks with a particular focus on the so called ``Dimpled Manifold Hypothesis'' by ~\citet{shamir2021dimpled}. Chapter one will cover the history and architecture of neural network architectures. Chapter two is focused on the background of adversarial vulnerability. Starting from the seminal paper by ~\citet{szegedy2013} we will develop the theory of adversarial perturbation and attack. Chapter three will build a theory of persistence that is related to Ricci Curvature, which can be used to measure properties of decision boundaries. We will use this foundation to make a conjecture relating adversarial attacks. Chapters four and five represent a sudden and wonderful digression that examines an intriguing related body of theory for spatial analysis of neural networks as approximations of kernel machines and becomes a novel theory for representing neural networks with bilinear maps. These heavily mathematical chapters will set up a framework and begin exploring applications of what may become a very important theoretical foundation for analyzing neural network learning with spatial and geometric information. We will conclude by setting up our new methods to address the conjecture from chapter 3 in continuing research.

There are a number of hypotheses underlying the existence of adversarial examples for classification problems. These include the high-dimensionality of the data, high codimension in the ambient space of the data manifolds of interest, and that the structure of machine learning models may encourage classifiers to develop decision boundaries close to data points. This article proposes a new framework for studying adversarial examples that does not depend directly on the distance to the decision boundary. Similarly to the smoothed classifier literature, we define a (natural or adversarial) data point to be $(\gamma,\sigma)$-stable if the probability of the same classification is at least $\gamma$ for points sampled in a Gaussian neighborhood of the point with a given standard deviation $\sigma$. We focus on studying the differences between persistence metrics along interpolants of natural and adversarial points. We show that adversarial examples have significantly lower persistence than natural examples for large neural networks in the context of the MNIST and ImageNet datasets. We connect this lack of persistence with decision boundary geometry by measuring angles of interpolants with respect to decision boundaries. Finally, we connect this approach with robustness by developing a manifold alignment gradient metric and demonstrating the increase in robustness that can be achieved when training with the addition of this metric.

Adversarial Training (AT) effectively improves the robustness of Deep Neural Networks (DNNs) to adversarial attacks. Generally, AT involves training DNN models with adversarial examples obtained within a pre-defined, fixed perturbation bound. Notably, individual natural examples from which these adversarial examples are crafted exhibit varying degrees of intrinsic vulnerabilities, and as such, crafting adversarial examples with fixed perturbation radius for all instances may not sufficiently unleash the potency of AT. Motivated by this observation, we propose two simple, computationally cheap vulnerability-aware reweighting functions for assigning perturbation bounds to adversarial examples used for AT, named Margin-Weighted Perturbation Budget (MWPB) and Standard-Deviation-Weighted Perturbation Budget (SDWPB). The proposed methods assign perturbation radii to individual adversarial samples based on the vulnerability of their corresponding natural examples. Experimental results show that the proposed methods yield genuine improvements in the robustness of AT algorithms against various adversarial attacks.

Deep neural networks are vulnerable to adversarial noise. Adversarial Training (AT) has been demonstrated to be the most effective defense strategy to protect neural networks from being fooled. However, we find AT omits to learning robust features, resulting in poor performance of adversarial robustness. To address this issue, we highlight two criteria of robust representation: (1) Exclusion: \emph{the feature of examples keeps away from that of other classes}; (2) Alignment: \emph{the feature of natural and corresponding adversarial examples is close to each other}. These motivate us to propose a generic framework of AT to gain robust representation, by the asymmetric negative contrast and reverse attention. Specifically, we design an asymmetric negative contrast based on predicted probabilities, to push away examples of different classes in the feature space. Moreover, we propose to weight feature by parameters of the linear classifier as the reverse attention, to obtain class-aware feature and pull close the feature of the same class. Empirical evaluations on three benchmark datasets show our methods greatly advance the robustness of AT and achieve state-of-the-art performance.

Adversarial Training (AT) has been found to substantially improve the robustness of deep learning classifiers against adversarial attacks. AT involves obtaining robustness by including adversarial examples in training a classifier. Most variants of AT algorithms treat every training example equally. However, recent works have shown that better performance is achievable by treating them unequally. In addition, it has been observed that AT exerts an uneven influence on different classes in a training set and unfairly hurts examples corresponding to classes that are inherently harder to classify. Consequently, various reweighting schemes have been proposed that assign unequal weights to robust losses of individual examples in a training set. In this work, we propose a novel instance-wise reweighting scheme. It considers the vulnerability of each natural example and the resulting information loss on its adversarial counterpart occasioned by adversarial attacks. Through extensive experiments, we show that our proposed method significantly improves over existing reweighting schemes, especially against strong white and black-box attacks.

Recent studies have shown that large pre-trained language models are vulnerable to adversarial attacks. Existing methods attempt to reconstruct the adversarial examples. However, these methods usually have limited performance in defense against adversarial examples, while also negatively impacting the performance on natural examples. To overcome this problem, we propose a method called Reactive Perturbation Defocusing (RPD). RPD uses an adversarial detector to identify adversarial examples and reduce false defenses on natural examples. Instead of reconstructing the adversaries, RPD injects safe perturbations into adversarial examples to distract the objective models from the malicious perturbations. Our experiments on three datasets, two objective models, and various adversarial attacks show that our proposed framework successfully repairs up to approximately 97% of correctly identified adversarial examples with only about a 2% performance decrease on natural examples. We also provide a demo of adversarial detection and repair based on our work.

Adversarial training (AT) is widely considered as the most promising strategy to defend against adversarial attacks and has drawn increasing interest from researchers. However, the existing AT methods still suffer from two challenges. First, they are unable to handle unrestricted adversarial examples (UAEs), which are built from scratch, as opposed to restricted adversarial examples (RAEs), which are created by adding perturbations bound by an $l_p$ norm to observed examples. Second, the existing AT methods often achieve adversarial robustness at the expense of standard generalizability (i.e., the accuracy on natural examples) because they make a tradeoff between them. To overcome these challenges, we propose a unique viewpoint that understands UAEs as imperceptibly perturbed unobserved examples. Also, we find that the tradeoff results from the separation of the distributions of adversarial examples and natural examples. Based on these ideas, we propose a novel AT approach called Provable Unrestricted Adversarial Training (PUAT), which can provide a target classifier with comprehensive adversarial robustness against both UAE and RAE, and simultaneously improve its standard generalizability. Particularly, PUAT utilizes partially labeled data to achieve effective UAE generation by accurately capturing the natural data distribution through a novel augmented triple-GAN. At the same time, PUAT extends the traditional AT by introducing the supervised loss of the target classifier into the adversarial loss and achieves the alignment between the UAE distribution, the natural data distribution, and the distribution learned by the classifier, with the collaboration of the augmented triple-GAN. Finally, the solid theoretical analysis and extensive experiments conducted on widely-used benchmarks demonstrate the superiority of PUAT.