Covariance matrices serve as compact descriptors of feature distributions in many machine-learning pipelines, including domain adaptation and Gaussian embeddings. Under a centered Gaussian approximation, the unregularized Wasserstein-2 optimal-transport (OT) discrepancy admits a closed form on covariances given by the Bures-Wasserstein (BW) objective on the symmetric positive definite (SPD) cone. We propose ITSPACE (Iterative Transport for Stable Proximal Alignment of Covariance Embeddings), a proximal majorization-minimization method that directly optimizes this exact BW objective through closed-form updates in a square-root factorization. In exact arithmetic, each iteration satisfies a sufficient-decrease inequality for the BW objective; under inexact polar computations, we provide an explicit certificate-gap bound controlling deviations from exact descent. The resulting iterations preserve PSD structure by construction and naturally support rank-restricted factors, making ITSPACE well-suited as a lightweight inner-loop primitive in settings where adaptation must be performed from unlabeled target batches under strict step and compute budgets. Across real-world covariance-alignment benchmarks, ITSPACE reaches low-BW-gap solutions substantially faster than BW-gradient descent, methods based on other covariance geometries, and entropically regularized sample-OT baselines.

Scientific discovery via symbolic regression is often viewed as statistically and computationally intractable because the hypothesis space of expressions grows combinatorially with depth. This paper revisits the statistical side through the lens of PAC learning, focusing on compositional function trees built from a finite vocabulary of smooth operators (e.g., $\{+,\times,\sin,\exp\}$ and affine maps). We prove that the relevant generalization quantity, Rademacher complexity, hence the excess risk, does not necessarily blow up exponentially with the number of distinct symbolic structures, but is controlled by (i) the depth $d$ and (ii) the Lipschitz constants of the base operators along the composed computation graph. Concretely, under mild Lipschitz conditions on operators and bounded affine leaves, a finite-union bound over a vocabulary of size $K=|\mathcal{H}_{\mathrm{base}}|$ together with Maurer-type vector contraction yields $\mathfrak{R}_n(\mathcal{H}_{\mathrm{comp}}^{d}) \leq (Kb\sqrt{2}L)^{d-1}\mathfrak{R}_n(\mathcal{H}_{\mathrm{comp}}^{1})$ with arity bound $b$; corresponding high-probability risk bounds scale as $\mathcal{O}(L^{d}/\sqrt{n})$ when $K,b=O(1)$ and $\mathfrak{R}_n(\mathcal{H}_{\mathrm{comp}}^{1})=O(n^{-1/2})$. We complement the theory with a modular codebase that trains differentiable operator trees (not MLPs) on synthetic "physics-like" targets of controlled depth and shows that the empirical generalization gap correlates positively with the predicted complexity term $(\widehat{L}^{d})/\sqrt{n}$.

We characterize weakly regularized continual classification in homogeneous models as sequential projections onto task margin sets. This result generalizes prior analyses restricted to either stationary (single-task) deep models or continual linear models. We show that global convergence generally fails, even for simple models linear in data but nonlinear in parameters. Nevertheless, by leveraging results from nonconvex projection theory, we identify regularity properties of homogeneous deep networks that guarantee local linear convergence under random and cyclic task sequences. Finally, we extend our analysis to continual regression, unifying the framework for homogeneous models.

AI agents are not your "coworkers" Marketing AI agents as digital employees may make human workers worse at spotting errors and more likely to offload accountability. Imagine coming in to work to learn that a new underling will report to you. The worker is not a person but an AI tool--one that your company nonetheless calls Alex, an "employee" with a title and defined responsibilities. How well do you think you would work with Alex? If you're anything like the managers recently studied by Emma Wiles, a Boston University business professor, treating Alex as a "coworker" and not a software tool would lead you to do a worse job. Wiles found that people caught 18% fewer errors when the work was said to have come from an agentic "AI employee" rather than a chatbot. It turns out that what's in a name matters.

With democracy on the decline in both the UK and US, Mehdi Hasan makes the case for independent journalism. Mehdi Hasan has had a front seat to US and UK politics for decades. With Britain facing yet another change in prime minister a decade after Brexit and the US looking ahead to its next vote with the midterm elections in November, we get his take on this moment and why independent journalism is needed more than ever. How did Colombia's election split a nation in two? Who's being left out of the World Cup? How is China using AI in the classroom?

In Russia, the impact of Ukraine's missile and drone strikes on energy infrastructure from Moscow to the Black Sea and beyond has long been evident. Drivers in the Russia-annexed Ukrainian peninsula of Crimea banned from filling their tanks so priority can be given to military vehicles. But such is the gravity of the situation it has now been explicitly acknowledged by President Vladimir Putin for the first time. Over the weekend, Russia's president discussed the crisis with senior officials and oil executives. And in public remarks, he was unusually frank. You're well aware that problems persist for both motorists and businesses, he told the meeting.

Plus: The US has allowed Anthropic to release Mythos 5 to "trusted" orgs. There are plenty of useful things a metric can reveal. There are even more that it can obscure or corrupt. Like a lot of people bitten by the self-quantifying bug, I started gathering personal data to pursue a nebulous collection of goals and desires. I wanted to feel better physically and emotionally, get outside more, and bring order to the messiness and uncertainty of my daily existence. But external metrics and data can never capture what's truly important.

Is the war entering a new phase? It was, perhaps, Ukraine's quietest victory over Russia's oldest and closest ally. Ukrainian President Volodymyr Zelenskyy urged neighbouring Belarus to shut down four Moscow-installed relay stations that help guide Russian drone attacks on Ukraine. The stations - originally cellular communication towers - relay signals for Russian drone operators and allow their unmanned aircraft to exchange information with each other and fly deep into western Ukraine, which has few drone interceptors and NATO-supplied air defence systems. The relayers did "make the signal stronger" and the Russian attacks "more precise", Andriy Pronin, one of the pioneers of drone warfare in Ukraine, told Al Jazeera.

Prime Minister Sanae Takaichi emphasized that the government will promote reforms of regulations and systems that fit the artificial intelligence era. The government's regulatory reform panel Monday called for easing building standards for data centers amid the rapid development of artificial intelligence. In response, Prime Minister Sanae Takaichi emphasized that the government will promote reforms of regulations and systems that fit the AI era. Lithium-ion batteries, crucial for data centers' stable operation, are regarded as hazardous materials under the fire service law and the building standards law, making it difficult to install them in sufficient numbers. The panel's proposal urged the government to exclude lithium-ion batteries from the restrictions by introducing safety standards for batteries. It also mentioned the type of AI that control robots that walk, seen as helpful in covering labor shortages in logistics, construction and elderly care services.

Swarms of cyborg insects controlled remotely via electrical implants can now operate underwater, thanks to tiny diving suits supplying them with oxygen - which could one day enable them to explore Mars. Hirotaka Sato at Nanyang Technological University in Singapore and his colleagues first demonstrated in 2021 that Madagascar hissing cockroaches () could be remotely controlled with electrodes embedded in sensory organs known as cerci. In 2024, they demonstrated that a swarm of 20 of these cyborg insects could coordinate. The aim was to develop biological robots equipped with infrared sensors that could be released in large numbers after natural disasters to search for survivors. Cockroaches represent a ready-made platform for such applications with a working fuel source, efficient locomotion and in-built reflexes to dodge obstacles - capabilities that engineers still struggle to replicate mechanically at such a small scale.