First, we define distributionally equivalent and Max-Information model extraction attacks, and reduce them into a variational optimisation problem.

Figure5: Detectedcausal pathsof the spreadof Covid-19amongthe federalGermanstates, including causes among the restriction measures taken by each federal state. Each colour (in arrows and policies) indicates causes of one state (see top legend). The four subfigures correspond to the four combinations of threshold 1 and 2 that we tested. A distribution P is faithful to a directed acyclic graph (DAG) G if no conditionalindependence relationsotherthanthe onesentailed by the Markov property are present. Let G be a causal graph with vertex setV and P be a probability distribution over the vertices inV generated by the causal structure represented by G. G and P satisfy the Causal Markov Condition if and only if for every W in V, W is independent of V\(Descendants(W) Parents(W)) given Parents(W).

Feature and Interaction Importance (FII) methods are essential in supervised learning for assessing the relevance of input variables and their interactions in complex prediction models. In many domains, such as personalized medicine, local interpretations for individual predictions are often required, rather than global scores summarizing overall feature importance. Random Forests (RFs) are widely used in these settings, and existing interpretability methods typically exploit tree structures and split statistics to provide model-specific insights. However, theoretical understanding of local FII methods for RF remains limited, making it unclear how to interpret high importance scores for individual predictions. We propose a novel, local, model-specific FII method that identifies frequent co-occurrences of features along decision paths, combining global patterns with those observed on paths specific to a given test point. We prove that our method consistently recovers the true local signal features and their interactions under a Locally Spike Sparse (LSS) model and also identifies whether large or small feature values drive a prediction. We illustrate the usefulness of our method and theoretical results through simulation studies and a real-world data example.

Pediatric appendicitis remains one of the most common causes of acute abdominal pain in children, and its diagnosis continues to challenge clinicians due to overlapping symptoms and variable imaging quality. This study aims to develop and evaluate a deep learning model based on a pretrained ResNet architecture for automated detection of appendicitis from ultrasound images. We used the Regensburg Pediatric Appendicitis Dataset, which includes ultrasound scans, laboratory data, and clinical scores from pediatric patients admitted with abdominal pain to Children Hospital. Hedwig in Regensburg, Germany. Each subject had 1 to 15 ultrasound views covering the right lower quadrant, appendix, lymph nodes, and related structures. For the image based classification task, ResNet was fine tuned to distinguish appendicitis from non-appendicitis cases. Images were preprocessed by normalization, resizing, and augmentation to enhance generalization. The proposed ResNet model achieved an overall accuracy of 93.44, precision of 91.53, and recall of 89.8, demonstrating strong performance in identifying appendicitis across heterogeneous ultrasound views. The model effectively learned discriminative spatial features, overcoming challenges posed by low contrast, speckle noise, and anatomical variability in pediatric imaging.

The proof, known to be so hard that a mathematician once offered 10 martinis to whoever could figure it out, uses number theory to explain quantum fractals. In 1974, five years before he wrote his Pulitzer Prize-winning book, Douglas Hofstadter was a graduate student in physics at the University of Oregon. When his doctoral adviser went on sabbatical to Regensburg, Germany, Hofstadter tagged along, hoping to practice his German. The pair joined a group of brilliant theoretical physicists who were agonizing over a particular problem in quantum theory. They wanted to determine the energy levels of an electron in a crystal grid placed near a magnet. Hofstadter was the odd one out, unable to follow the others' line of thought. "Part of my luck was that I couldn't keep up with them," he said.

We study the problem of constructing coresets for $(k, z)$-clustering when the input dataset is corrupted by stochastic noise drawn from a known distribution. In this setting, evaluating the quality of a coreset is inherently challenging, as the true underlying dataset is unobserved. To address this, we investigate coreset construction using surrogate error metrics that are tractable and provably related to the true clustering cost. We analyze a traditional metric from prior work and introduce a new error metric that more closely aligns with the true cost. Although our metric is defined independently of the noise distribution, it enables approximation guarantees that scale with the noise level. We design a coreset construction algorithm based on this metric and show that, under mild assumptions on the data and noise, enforcing an $\varepsilon$-bound under our metric yields smaller coresets and tighter guarantees on the true clustering cost than those obtained via classical metrics. In particular, we prove that the coreset size can improve by a factor of up to $\mathrm{poly}(k)$, where $n$ is the dataset size. Experiments on real-world datasets support our theoretical findings and demonstrate the practical advantages of our approach.