Characterizing precisely the asymptotic generalization error of neural networks using parameters that can be estimated efficiently is a crucial problem in machine learning, which relies heavily on heuristics and practitioners' intuition to make key design choices. In order to mitigate this issue, we introduce the Representation Gap, a metric closely related to the generalization error, but admitting better-behaved asymptotic dynamics. Focusing on equivariant diffusion models and leveraging results from optimal quantization and point-process theory, we derive a precise asymptotic equivalent of the Representation Gap and show that it is governed by a single parameter, the \textit{intrinsic dimension} of the task, which is easy to interpret, efficient to estimate, and can be linked to the equivariances of common neural network architectures. We show that this asymptotic dynamic also extends to a broader range of tasks and training algorithms. Finally, we demonstrate empirically that our asymptotic law and intrinsic dimension estimation are accurate on a wide range of synthetic datasets, where these quantities are known, as well as on more realistic datasets, where we obtain results consistent with the related literature.

Air France and Airbus have been found guilty of manslaughter over a 2009 plane crash which killed 228 people. The Paris Appeals Court found the airline and aircraft manufacturer guilty of corporate manslaughter over the incident, in which flight AF447 between Rio de Janeiro and Paris crashed into the Atlantic Ocean. The passenger jet stalled during a storm and plunged into the water, killing all on board. A court had previously cleared the companies in April 2023 but they were found guilty after this appeal. The Airbus A330 vanished from radars during a storm, with its wreckage found after a long search of 10,000 sq km (3,860 sq miles) of sea floor.

Accurate trend forecasting in healthcare time series is essential for planning and resource allocation. This paper proposes a Bayesian framework for predicting oncology demand trends, modeling weekly appointments as a Poisson process with a Gamma prior to the demand rate. To enhance adaptability and capture persistent directional patterns, we incorporate a residual-based boosting mechanism grounded in a Gamma-Log-Normal conjugate structure. This boosting approach allows the model to track both short- and long-term trend shifts while maintaining the analytical tractability of conjugate Bayesian updating. The methodology was evaluated on real oncology service data from Cariri, Ceara, Brazil, and compared against established baselines, including linear regression, ARIMA, naive forecasting, LSTM neural networks, and XGBoost. Results showed that the proposed model outperforms competing methods in trend detection accuracy, with gains in terms of percentage of correct direction of 38.25% in relation to the second best approach in some cases.

This paper introduces the use of per-class regularization hyperparameters in Gabriel graph-based binary classifiers. We demonstrate how the quality index used for regularization behaves both in the margin region and in the presence of outliers, and how incorporating this regularization flexibility can lead to solutions that effectively eliminate outliers while training the classifier. We also show how it can address class imbalance by generating higher and lower thresholds for the majority and minority classes, respectively. Thus, rather than having a single solution based on fixed thresholds, flexible thresholds expand the solution space and can be optimized through hyperparameter tuning algorithms. Friedman test shows that flexible thresholds are capable of improving Gabriel graph-based classifiers.

Learning low-dimensional representations from multi-view relational data is challenging when underlying geometries differ across views. We propose Bary-GWMDS, a Gromov-Wasserstein-based method that operates directly on distance matrices to learn a consensus embedding preserving shared relational structure. By leveraging intrinsic distances, the approach naturally handles nonlinear distortions across views. We also introduce Mean-GWMDS-C, a clustering-oriented formulation that averages distance matrices and learns reduced-support representations via a consensus Gromov-Wasserstein transport. Experiments on synthetic and real-world datasets show that the proposed framework yields stable and geometrically meaningful embeddings.

The recent surge in fuel prices due to the war in Iran has spurred demand for electric vehicles around the world, and Chinese car makers are making the most of the opportunity. China is the world's top producer of EVs, and while its manufacturers remain largely shut out of the major car market of the United States, they are benefiting from an uptick in interest and orders via dealerships across Asia and elsewhere. BYD, which overtook Tesla as the world's largest seller of electric vehicles last year and is expanding aggressively overseas, is at the centre of this shift in focus. We survive and are successful without the US market today, BYD executive vice president Stella Li told the BBC at the Beijing Auto Show. Instead of aiming for US customers, the company says its challenge is meeting increased demand in other regions, including Brazil, the UK and Europe.

What does the data tell us about immigration in Wales? Like many countries, Wales sees a steady flow of people arriving and leaving for other countries each year. The difference between those arriving and those leaving is known as net migration. Focusing on people moving from abroad, latest estimates say Wales' population - which was 3.2 million in June 2024 - had increased by about 23,000 over the previous year as a result of net international migration. A recent YouGov poll found a quarter of people surveyed in Wales believed that immigration, alongside the economy, should be among the issues prioritised by the Welsh government, even though immigration is controlled by the UK government.

Near the Kremlin several dozen people are queuing outside the presidential administration office. They've come to submit petitions calling on President Vladimir Putin to end a crackdown on the internet. Russian authorities have been tightening control of the country's cyber space. Access to global messaging apps has been restricted and there are widespread disruptions to, even shutdowns of, mobile internet. Petitioning the president is legal.

A nature-loving 10-year-old girl who found an endangered amphibian under a bridge has left her mum in shock, surprise and disbelief. Melanie Hill said her daughter, Evie, discovered the nine-inch Mexican axolotl as they spent the day near the River Ogmore in Bridgend. She said Evie was always finding things like newts and bugs, but said the axolotl discovery was a surprise. It is the first documented discovery of an axolotl in the wild in the UK with only 50 to 1,000 individuals left globally today, according to experts. Axolotls as pets have seen a surge in popularity in recent years after they were introduced to video games such as Minecraft and Roblox.

McDonald's boss on abuse claims: 'I don't want to talk about the past' The boss of McDonald's UK and Ireland has said she doesn't want to talk about the past when asked about allegations of abuse at the fast-food chain. Lauren Schultz told the BBC what had happened in recent years was unacceptable but said we have drawn a line under it. A BBC investigation in 2023 heard from more than 100 McDonald's workers in the UK claiming they faced a toxic culture of sexual assault, harassment, racism, and bullying. Last year, staff said they still faced sexual abuse and harassment. The UK equality watchdog agreed tougher measures with the company to protect staff in November, including new sexual harassment training.