Any continuous function $f^*$ can be approximated arbitrarily well by a neural network with sufficiently many neurons $k$. We consider the case when $f^*$ itself is a neural network with one hidden layer and $k$ neurons. Approximating $f^*$ with a neural network with $n< k$ neurons can thus be seen as fitting an under-parameterized "student" network with $n$ neurons to a "teacher" network with $k$ neurons. As the student has fewer neurons than the teacher, it is unclear, whether each of the $n$ student neurons should copy one of the teacher neurons or rather average a group of teacher neurons. For shallow neural networks with erf activation function and for the standard Gaussian input distribution, we prove that "copy-average" configurations are critical points if the teacher's incoming vectors are orthonormal and its outgoing weights are unitary. Moreover, the optimum among such configurations is reached when $n-1$ student neurons each copy one teacher neuron and the $n$-th student neuron averages the remaining $k-n+1$ teacher neurons. For the student network with $n=1$ neuron, we provide additionally a closed-form solution of the non-trivial critical point(s) for commonly used activation functions through solving an equivalent constrained optimization problem. Empirically, we find for the erf activation function that gradient flow converges either to the optimal copy-average critical point or to another point where each student neuron approximately copies a different teacher neuron. Finally, we find similar results for the ReLU activation function, suggesting that the optimal solution of underparameterized networks has a universal structure.

The Strait of Gibraltar is a region characterized by intricate oceanic sub-mesoscale features, influenced by topography, tidal forces, instabilities, and nonlinear hydraulic processes, all governed by the nonlinear equations of fluid motion. In this study, we aim to uncover the underlying physics of these phenomena within 3D MIT general circulation model simulations, including waves, eddies, and gyres. To achieve this, we employ Dynamic Mode Decomposition (DMD) to break down simulation snapshots into Koopman modes, with distinct exponential growth/decay rates and oscillation frequencies. Our objectives encompass evaluating DMD's efficacy in capturing known features, unveiling new elements, ranking modes, and exploring order reduction. We also introduce modifications to enhance DMD's robustness, numerical accuracy, and robustness of eigenvalues. DMD analysis yields a comprehensive understanding of flow patterns, internal wave formation, and the dynamics of the Strait of Gibraltar, its meandering behaviors, and the formation of a secondary gyre, notably the Western Alboran Gyre, as well as the propagation of Kelvin and coastal-trapped waves along the African coast. In doing so, it significantly advances our comprehension of intricate oceanographic phenomena and underscores the immense utility of DMD as an analytical tool for such complex datasets, suggesting that DMD could serve as a valuable addition to the toolkit of oceanographers.

In recent years, there has been considerable interest in developing machine learning models on graphs in order to account for topological inductive biases. In particular, recent attention was given to Gaussian processes on such structures since they can additionally account for uncertainty. However, graphs are limited to modelling relations between two vertices. In this paper, we go beyond this dyadic setting and consider polyadic relations that include interactions between vertices, edges and one of their generalisations, known as cells. Specifically, we propose Gaussian processes on cellular complexes, a generalisation of graphs that captures interactions between these higher-order cells. One of our key contributions is the derivation of two novel kernels, one that generalises the graph Mat\'ern kernel and one that additionally mixes information of different cell types.

We study gradient flow on the multi-index regression problem for high-dimensional Gaussian data. Multi-index functions consist of a composition of an unknown low-rank linear projection and an arbitrary unknown, low-dimensional link function. As such, they constitute a natural template for feature learning in neural networks. We consider a two-timescale algorithm, whereby the low-dimensional link function is learnt with a non-parametric model infinitely faster than the subspace parametrizing the low-rank projection. By appropriately exploiting the matrix semigroup structure arising over the subspace correlation matrices, we establish global convergence of the resulting Grassmannian population gradient flow dynamics, and provide a quantitative description of its associated `saddle-to-saddle' dynamics. Notably, the timescales associated with each saddle can be explicitly characterized in terms of an appropriate Hermite decomposition of the target link function. In contrast with these positive results, we also show that the related \emph{planted} problem, where the link function is known and fixed, in fact has a rough optimization landscape, in which gradient flow dynamics might get trapped with high probability.

Yemen's Houthi militia claimed an attempted attack on southern Israel on Tuesday, saying it had launched a "large batch" of ballistic and cruise missiles as well as drones toward Israeli targets. The Iran-backed militia carried out the attempted assault in response to what it called "brutal Israeli-American aggression" in Gaza, the Houthi military spokesman, Yahya Sarea, said on the social media platform X. Mr. Sarea said the attack was the third operation conducted by the Houthis "in support of our persecuted brothers in Palestine," and threatened further missile and drone assaults. The Times could not independently verify the Houthi claims. On Tuesday, the Israeli military said its aerial defense system had intercepted a surface-to-surface missile fired toward Israel "from the area of the Red Sea."

Who is the more trustworthy custodian of machines capable of diverting the course of human civilisation: Elon Musk or the Chinese Communist party? Will it be the billionaire megalomaniac tycoon who meddles in international crises as if they were video games loaded on to his personal propaganda console? Or the authoritarian superpower that likes digital technology best when it enables more efficient and ruthless social engineering and political repression? Neither is the answer – and thankfully, other options are available. But asking the question in starkly polarised terms illuminates the challenge posed by artificial intelligence that is evolving faster than any effort to bring it under responsible supervision. Everyone can agree that there should be rules because nobody wants awesome computational capability to fall into the wrong hands.

What U.K. Prime Minister Rishi Sunak couldn't get from his fellow world leaders, he might get from Elon Musk. The American tech billionaire will bring some star power to Sunak's summit on AI safety this week, including a conversation between the two men live-streamed on Musk's social media platform, X. Sunak organized the international gathering to reassert Britain's influence in the wake of Brexit and gain an early advantage in a potentially era-defining technology. While the event at Bletchley Park, the home of Britain's World War II code-breakers, has attracted numerous top tech executives, state leaders have been harder to come by, as many focus on the threat of an expanding conflict in the Middle East. Italian premier Giorgia Meloni is expected to be the only leader of a Group of Seven nation to attend, with Vice President Kamala Harris representing the U.S.

Nick Clegg has compared the clamour over artificial intelligence to the 80s-era "moral panic" over video games, firing a warning shot to international politicians and regulators as they gather for a two-day summit on AI safety. The former UK deputy prime minister who is now president of global affairs at Mark Zuckerberg's Meta said AI was caught in a "great hype cycle" but warned that new technologies inspired a mixture of excessive zeal and excessive pessimism. British officials are hoping to use the summit, which starts on Wednesday at Bletchley Park, to kickstart a regulatory process that could mirror international attempts to combat the climate crisis. But Clegg's comments show they are likely to encounter resistance from some of the industry's most powerful companies. "New technologies always lead to hype," he said.

Recent language models have been improved by the addition of external memory. Nearest neighbor language models retrieve similar contexts to assist in word prediction. The addition of locality levels allows a model to learn how to weight neighbors based on their relative location to the current text in source documents, and have been shown to further improve model performance. Nearest neighbor models have been explored for controllable generation but have not examined the use of locality levels. We present a novel approach for this purpose and evaluate it using automatic and human evaluation on politeness, formality, supportiveness, and toxicity textual data. We find that our model is successfully able to control style and provides a better fluency-style trade-off than previous work.

This paper introduces the Gaussian multi-Graphical Model, a model to construct sparse graph representations of matrix- and tensor-variate data. We generalize prior work in this area by simultaneously learning this representation across several tensors that share axes, which is necessary to allow the analysis of multimodal datasets such as those encountered in multi-omics. Our algorithm uses only a single eigendecomposition per axis, achieving an order of magnitude speedup over prior work in the ungeneralized case. This allows the use of our methodology on large multi-modal datasets such as single-cell multi-omics data, which was challenging with previous approaches. We validate our model on synthetic data and five real-world datasets.