Holger Rauhut

Vinh Hoang, Sebastian Krumscheid, Holger Rauhut et al.

We propose a novel deterministic purification method to improve adversarial robustness by mapping a potentially adversarial sample toward a nearby sample that lies close to a mode of the data distribution, where classifiers are more reliable. We design the method to be deterministic to ensure reliable test accuracy and to prevent the degradation of effective robustness observed in stochastic purification approaches when the adversary has full knowledge of the system and its randomness. We employ a score model trained by minimizing the expected reconstruction error of noise-corrupted data, thereby learning the structural characteristics of the input data distribution. Given a potentially adversarial input, the method searches within its local neighborhood for a purified sample that minimizes the expected reconstruction error under noise corruption and then feeds this purified sample to the classifier. During purification, sharpness-aware minimization is used to guide the purified samples toward flat regions of the expected reconstruction error landscape, thereby enhancing robustness. We further show that, as the noise level decreases, minimizing the expected reconstruction error biases the purified sample toward local maximizers of the Gaussian-smoothed density; under additional local assumptions on the score model, we prove recovery of a local maximizer in the small-noise limit. Experimental results demonstrate significant gains in adversarial robustness over state-of-the-art methods under strong deterministic white-box attacks.

Jona-Maria Diederen, Holger Rauhut, Ulrich Terstiege

Convolutional neural networks are widely used in imaging and image recognition. Learning such networks from training data leads to the minimization of a non-convex function. This makes the analysis of standard optimization methods such as variants of (stochastic) gradient descent challenging. In this article we study the simplified setting of linear convolutional networks. We show that the gradient flow (to be interpreted as an abstraction of gradient descent) applied to the empirical risk defined via certain loss functions including the square loss always converges to a critical point, under a mild condition on the training data.

Laura Paul, Holger Rauhut, Martin Burger et al.

Recent advances in imaging technologies, deep learning and numerical performance have enabled non-invasive detailed analysis of artworks, supporting their documentation and conservation. In particular, automated detection of craquelure in digitized paintings is crucial for assessing degradation and guiding restoration, yet remains challenging due to the possibly complex scenery and the visual similarity between cracks and crack-like artistic features such as brush strokes or hair. We propose a hybrid approach that models crack detection as an inverse problem, decomposing an observed image into a crack-free painting and a crack component. A deep generative model is employed as powerful prior for the underlying artwork, while crack structures are captured using a Mumford--Shah-type variational functional together with a crack prior. Joint optimization yields a pixel-level map of crack localizations in the painting.