Important research efforts have focused on the design and training of neural networks with a controlled Lipschitz constant. The goal is to increase and sometimes guarantee the robustness against adversarial attacks. Recent promising techniques draw inspirations from different backgrounds to design 1-Lipschitz neural networks, just to name a few: convex potential layers derive from the discretization of continuous dynamical systems, Almost-Orthogonal-Layer proposes a tailored method for matrix rescaling. However, it is today important to consider the recent and promising contributions in the field under a common theoretical lens to better design new and improved layers. This paper introduces a novel algebraic perspective unifying various types of 1-Lipschitz neural networks, including the ones previously mentioned, along with methods based on orthogonality and spectral methods. Interestingly, we show that many existing techniques can be derived and generalized via finding analytical solutions of a common semidefinite programming (SDP) condition. We also prove that AOL biases the scaled weight to the ones which are close to the set of orthogonal matrices in a certain mathematical manner. Moreover, our algebraic condition, combined with the Gershgorin circle theorem, readily leads to new and diverse parameterizations for 1-Lipschitz network layers. Our approach, called SDP-based Lipschitz Layers (SLL), allows us to design non-trivial yet efficient generalization of convex potential layers. Finally, the comprehensive set of experiments on image classification shows that SLLs outperform previous approaches on certified robust accuracy. Code is available at https://github.com/araujoalexandre/Lipschitz-SLL-Networks.
Crowd counting has drawn much attention due to its importance in safety-critical surveillance systems. Especially, deep neural network (DNN) methods have significantly reduced estimation errors for crowd counting missions. Recent studies have demonstrated that DNNs are vulnerable to adversarial attacks, i.e., normal images with human-imperceptible perturbations could mislead DNNs to make false predictions. In this work, we propose a robust attack strategy called Adversarial Patch Attack with Momentum (APAM) to systematically evaluate the robustness of crowd counting models, where the attacker's goal is to create an adversarial perturbation that severely degrades their performances, thus leading to public safety accidents (e.g., stampede accidents). Especially, the proposed attack leverages the extreme-density background information of input images to generate robust adversarial patches via a series of transformations (e.g., interpolation, rotation, etc.). We observe that by perturbing less than 6\% of image pixels, our attacks severely degrade the performance of crowd counting systems, both digitally and physically. To better enhance the adversarial robustness of crowd counting models, we propose the first regression model-based Randomized Ablation (RA), which is more sufficient than Adversarial Training (ADT) (Mean Absolute Error of RA is 5 lower than ADT on clean samples and 30 lower than ADT on adversarial examples). Extensive experiments on five crowd counting models demonstrate the effectiveness and generality of the proposed method. The supplementary materials and certificate retrained models are available at \url{https://www.dropbox.com/s/hc4fdx133vht0qb/ACM_MM2021_Supp.pdf?dl=0}
We extend randomized smoothing to cover parameterized transformations (e.g., rotations, translations) and certify robustness in the parameter space (e.g., rotation angle). This is particularly challenging as interpolation and rounding effects mean that image transformations do not compose, in turn preventing direct certification of the perturbed image (unlike certification with $\ell^p$ norms). We address this challenge by introducing three different kinds of defenses, each with a different guarantee (heuristic, distributional and individual) stemming from the method used to bound the interpolation error. Importantly, we show how individual certificates can be obtained via either statistical error bounds or efficient online inverse computation of the image transformation. We provide an implementation of all methods at https://github.com/eth-sri/transformation-smoothing.