This results in the large dips in values we observe in the learned extended values.

Germany's Munich airport has resumed operations after drone sightings led to the cancellation of 17 flights, the diversion of 15 others and the stranding of some 3,000 passengers. Flights had restarted by early Friday, with flight tracking websites showing planes departing the airport at about 5:50am (03:50 GMT). At least 19 Lufthansa flights were affected, either cancelled or re-routed, because of the airport suspension, the spokesperson added. Earlier, the airport said that drone sightings were first reported by German air traffic control at 10:18pm local time [20:18 GMT] on Thursday, leading initially to a restriction on flights, which was then upgraded to a full suspension. Germany's DPA news agency said police reported that several people had seen a drone near the airport, with later sightings of a drone over the airport grounds.

We present a technique for significantly speeding up Alternating Least Squares (ALS) and Gradient Descent (GD), two widely used algorithms for tensor factorization. By exploiting properties of the Khatri-Rao product, we show how to efficiently address a computationally challenging sub-step of both algorithms. Our algorithm, DFacTo, only requires two sparse matrix-vector products and is easy to parallelize. DFacTo is not only scalable but also on average 4 to 10 times faster than competing algorithms on a variety of datasets. For instance, DFacTo only takes 480 seconds on 4 machines to perform one iteration of the ALS algorithm and 1,143 seconds to perform one iteration of the GD algorithm on a 6.5 million 2.5 million 1.5 million dimensional tensor with 1.2 billion non-zero entries.

Germany's Munich airport has reopened after several drone sightings forced it to close and cancel more than a dozen flights on Thursday night. At least 17 flights were grounded in Munich, affecting nearly 3,000 passengers, while the airport said it diverted a further 15 flights to nearby cities. On Friday, a spokesperson for German flag carrier Lufthansa said flight operations have since resumed according to schedule. There was no immediate confirmation of where the drones had come from. Several airports across Europe have closed down in recent weeks because of unidentified drones.

The training dynamics of neural networks have attracted significant attention in deep learning theory. It has been suggested that the dynamics induced by training algorithms strongly influence the generalization performance of neural networks. This effect is captured in the idea of implicit bias (Neyshabur et al., 2014), in which the algorithm selects a certain solution among many induced by nonconvexity of the loss and overparametrization of networks. Accordingly, many recent works have studied the interplay between models and optimizers, aiming to characterize the resulting implicit biases (Neyshabur, 2017; Soudry et al., 2018; Arora et al., 2019; Bartlett et al., 2021). Moreover, understanding the convergence speed and timescales of the training dynamics contributes to efficient training of high-performance models in practice, especially in the context of modern large-scale neural networks in which the training is stopped at a compute-optimal point (Kaplan et al., 2020).

This paper analyzes the generalization error of minimum-norm interpolating solutions in linear regression using spiked covariance data models. The paper characterizes how varying spike strengths and target-spike alignments can affect risk, especially in overparameterized settings. The study presents an exact expression for the generalization error, leading to a comprehensive classification of benign, tempered, and catastrophic overfitting regimes based on spike strength, the aspect ratio $c=d/n$ (particularly as $c \to \infty$), and target alignment. Notably, in well-specified aligned problems, increasing spike strength can surprisingly induce catastrophic overfitting before achieving benign overfitting. The paper also reveals that target-spike alignment is not always advantageous, identifying specific, sometimes counterintuitive, conditions for its benefit or detriment. Alignment with the spike being detrimental is empirically demonstrated to persist in nonlinear models.

Standard discrete diffusion models treat all unobserved states identically by mapping them to an absorbing [MASK] token. This creates an 'information void' where semantic information that could be inferred from unmasked tokens is lost between denoising steps. We introduce Continuously Augmented Discrete Diffusion (CADD), a framework that augments the discrete state space with a paired diffusion in a continuous latent space. This yields graded, gradually corrupted states in which masked tokens are represented by noisy yet informative latent vectors rather than collapsed 'information voids'. At each reverse step, CADD may leverage the continuous latent as a semantic hint to guide discrete denoising. The design is clean and compatible with existing discrete diffusion training. At sampling time, the strength and choice of estimator for the continuous latent vector enables a controlled trade-off between mode-coverage (generating diverse outputs) and mode-seeking (generating contextually precise outputs) behaviors. Empirically, we demonstrate CADD improves generative quality over mask-based diffusion across text generation, image synthesis, and code modeling, with consistent gains on both qualitative and quantitative metrics against strong discrete baselines.

Linking textual values in tabular data to their corresponding entities in a Knowledge Base is a core task across a variety of data integration and enrichment applications. Although Large Language Models (LLMs) have shown State-of-The-Art performance in Entity Linking (EL) tasks, their deployment in real-world scenarios requires not only accurate predictions but also reliable uncertainty estimates, which require resource-demanding multi-shot inference, posing serious limits to their actual applicability. As a more efficient alternative, we investigate a self-supervised approach for estimating uncertainty from single-shot LLM outputs using token-level features, reducing the need for multiple generations. Evaluation is performed on an EL task on tabular data across multiple LLMs, showing that the resulting uncertainty estimates are highly effective in detecting low-accuracy outputs. This is achieved at a fraction of the computational cost, ultimately supporting a cost-effective integration of uncertainty measures into LLM-based EL workflows. The method offers a practical way to incorporate uncertainty estimation into EL workflows with limited computational overhead.