Quantum-Inspired Analysis of Neural Network Vulnerabilities: The Role of Conjugate Variables in System Attacks

TL;DR

Neural networks are shown to possess intrinsic vulnerabilities to imperceptible adversarial perturbations. The authors introduce a quantum-inspired framework that treats input features as $\hat{x}_i$ and loss-gradients as conjugate attack operators $\hat{p}_i=\frac{\partial}{\partial x_i}$, yielding an uncertainty bound $\Delta x_i \Delta p_i \ge \frac{1}{2}$. Empirical results across MNIST and CIFAR-10 demonstrate the existence of a fundamental accuracy–robustness trade-off consistent with the bound, with feature-space attacks proving more effective than pixel-space attacks. The work highlights that neural networks can be analyzed as complex physical systems, offering physics-inspired insights for robustness and design that span interdisciplinary boundaries and potentially guide future robustness-enhancement strategies.

Abstract

Neural networks demonstrate inherent vulnerability to small, non-random perturbations, emerging as adversarial attacks. Such attacks, born from the gradient of the loss function relative to the input, are discerned as input conjugates, revealing a systemic fragility within the network structure. Intriguingly, a mathematical congruence manifests between this mechanism and the quantum physics' uncertainty principle, casting light on a hitherto unanticipated interdisciplinarity. This inherent susceptibility within neural network systems is generally intrinsic, highlighting not only the innate vulnerability of these networks but also suggesting potential advancements in the interdisciplinary area for understanding these black-box networks.

Figures (2)

• Figure 1: Illustration of $\Delta x$ and $\Delta p$ in a three-layer convolutional neural network trained on the MNIST dataset over 50 epochs. The data's high-dimensional feature space was reduced to two dimensions using the t-SNE (t-Distributed Stochastic Neighbor Embedding) algorithm for easy visualization. (A) Shaded regions indicate the class predictions obtained by the finally trained network, and the colors imposed on individual points indicate the true labels of corresponding test samples. (B) All test samples were subjected to the Projected Gradient Descent (PDG) adversarial attack method kurakin2017adversarialmadry2018towards with $\epsilon=0.1$ and $\alpha=0.1/4$ over four iterative steps. It is seen that these adversarially perturbed samples are evidently deviated from class regions they should be located. (C) The prediction region evolution for the digit '8' is displayed at epochs 1, 21, and 41. More deeper the color is, more confident the prediction is by the network. (D) The shaded area is similar to (C), but with points representing the adversarial predictions of the attacked images, illustrating the temporal impact of the PDG attack on model accuracy.
• Figure 2: Results of the three different types of neural networks: a three-layer convolutional network running on the MNIST dataset, a four-layer convolutional network on the CIFAR-10 dataset, and a residual network7780459 with eight convolutional layers on the CIFAR-10 dataset. The term "feature" in the labels represents the results obtained by attacking the features of the input images, while "pixel" corresponds to attacks directed at the pixels themselves. Each neural network underwent training for a span of 50 epochs. The quantities $\Delta x$ and $\Delta p$ were determined through high-dimensional Monte-Carlo integrations. Subfigures (A), (C), (E), (G), (I), and (K) depict the test and robust accuracy metrics, with the robust accuracy evaluated on images perturbed by the PDG adversarial attack method, using parameters $\epsilon=8/255$ and $\alpha=2/255$ across four iterative steps. Subfigures (B), (D), (F), (H), (J), and (L) illustrate the trade-off relationship between $\Delta x$ and $\Delta p$.