Riemannian neural networks have proven effective in solving a variety of machine learning tasks. The key to their success lies in the development of principled Riemannian analogs of fundamental building blocks in deep neural networks (DNNs). Among those, Riemannian batch normalization (BN) layers have shown to enhance training stability and improve accuracy. In this paper, we propose BN layers for neural networks on complex domains. The proposed layers have close connections with existing Riemannian BN layers. We derive essential components for practical implementations of BN layers on some complex domains which are less studied in previous works, e.g., the Siegel disk domain. We conduct experiments on radar clutter classification, node classification, and action recognition demonstrating the efficacy of our method.

Gradient Boosting Decision Trees (GBDTs) dominate tabular machine learning, with modern implementations like XGBoost, LightGBM, and CatBoost being based on Newton boosting: a second-order descent step in the space of decision trees. Despite its empirical success, the global convergence of Newton boosting is poorly understood compared to first-order boosting. In this paper, we introduce Restricted Newton Descent, which studies convex optimization with Newton's method on Hilbert spaces with inexact iterates, based on the concepts of cosine angle and weak gradient edge. Within this framework, we recover Newton boosting with GBDTs and classical finite-dimensional theory as special cases. We first prove that vanilla Newton boosting achieves a linear rate of convergence for smooth, strongly convex losses that satisfy a Hessian-dominance condition. To handle general convex losses with Lipschitz Hessians, we extend a recent gradient regularized Newton scheme to the restricted weak learner setting. This scheme minimally modifies the classical algorithm by introducing an adaptive $\ell_2$-regularization term proportional to the square root of the gradient norm at each iteration. We establish a $\mathcal{O}(\frac{1}{k^2})$ rate for this scheme, thereby obtaining a globally convergent second-order GBDT algorithm with a rate matching that of first-order boosting with Nesterov momentum. In numerical experiments, we show that our scheme converges while vanilla Newton boosting may diverge.

Decentralized learning is a learning process in which data is distributed across computational agents or collected by individual agents, and model parameters are computed as the consensus of the agents. It has gained a lot of interest for applications where agents can collaboratively learn a predictive model without sharing their own data, but sharing only their local models with their immediate neighbors to generate a global model [He et al., 2018, Hendrikx et al., 2019, Arjevani et al., 2020]. We assume there are N agents who are connected over an undirected communication network G = (V,E) where V = {1,...,N} represents the agents and E V V denotes the set of edges; i.e., if agent i and j are connected then (i,j) E implies (j,i) E. Suppose we have a collection of n independent and identically distributed (i.i.d.) data pairs zi = (ai,yi), where ai Rp is the feature vector and yi the label or response of the i-th observation. Let Z = [z1,z2,,zn] Rnp be sampled from the distribution p(Z|x) where the parameter x Rd has a common prior. The goal is to sample from the posterior distribution p(x|Z) p(Z|x)p(x) by distributing Z among N agents such that Zi = {zi1,zi2,,zini} is the subset of data exclusive to agent i.

Randomized-subspace methods reduce the cost of first-order optimization by using only low-dimensional projected-gradient information, a feature that is attractive in forward-mode automatic differentiation and communication-limited settings. While Nesterov acceleration is well understood for full-gradient and coordinate-based methods, obtaining accelerated methods for general subspace sketches that use only projected-gradient information and can improve over full-dimensional Nesterov acceleration in oracle complexity is technically nontrivial. We develop randomized-subspace Nesterov accelerated gradient methods for smooth convex and smooth strongly convex optimization under matrix smoothness and generic sketch moment assumptions. The key technical ingredient is a three-sequence formulation tailored to matrix smoothness, which recovers the corresponding classical Nesterov methods in the full-dimensional case. The resulting theory establishes accelerated oracle-complexity guarantees and makes explicit how matrix smoothness and the sketch distribution enter the complexity. It also provides a unified basis for comparing sketch families and identifying when randomized-subspace acceleration improves over full-dimensional Nesterov acceleration in oracle complexity.

We study recursive maximum likelihood estimation for stochastic interacting particle systems based on continuous observation of a single particle. In this regime, consistent estimation of the finite-particle log-likelihood is not possible, even in the limit as the number of particles $N\rightarrow\infty$ and the time horizon $t\rightarrow\infty$. We thus seek to optimise the stationary log-likelihood of the limiting mean-field system. We achieve this via a form of stochastic gradient estimate in continuous time, with stochastic gradient estimates computed using the continuous trajectory of the single observed particle, alongside a virtual interacting particle system and a virtual tangent interacting particle system, which are integrated with the online parameter estimate. For fixed numbers of real and virtual particles, we show that the resulting algorithms drive the gradient of a finite-particle surrogate objective to zero as $t\to\infty$. We then prove that, in the iterated limit $t\to\infty$ followed by $N,M\to\infty$, these surrogate gradients converge uniformly to the gradient of the stationary log-likelihood of the limiting mean-field system, yielding convergence to its stationary points. We illustrate the method on several numerical examples, including a model with quadratic confinement and interaction potentials, a model of interacting FitzHugh--Nagumo neurons, and a stochastic Kuramoto model.

The UAE says it will integrate agentic AI across half of its government operations within two years, making one of the most aggressive moves in the global AI race.

How does live facial recognition work and how many police forces use it? Britain's biometrics watchdogs have warned that national oversight of AI-powered face scanning to catch criminals is lagging far behind the technology's rapid growth. With the Metropolitan police almost doubling the number of faces they scan in London over the past 12 months and a rising use of the technology by retailers in the UK, Prof William Webster, the biometrics commissioner for England and Wales, said the "slow pace of legislation was trying to catch up with the real world" and "the horse had gone before the cart". Dr Brian Plastow, who holds the same role in Scotland, warned the technology was "nowhere near as effective as the police claim it is" and said there was a "patchwork legal framework" throughout the UK. He said in England and Wales, police were "really just marking their own homework".

Varun Chandra advises Keir Starmer on trade negotiations including AI investment. Varun Chandra advises Keir Starmer on trade negotiations including AI investment. Exclusive: Varun Chandra's talks with Google, Meta, Apple and others raise fears of'lobbying behind closed doors' An influential government adviser close to Keir Starmer and Rachel Reeves held 16 undisclosed meetings with top US tech executives, the Guardian can reveal. The No 10 business aide Varun Chandra discussed regulatory changes, AI and Donald Trump's second administration with tech corporations during confidential meetings between October 2024 and October 2025. In one meeting he offered to help a top executive meet the prime minister directly.

Things to Do in L.A. Tap to enable a layout that focuses on the article. Cole Allen's journey from young athlete and Caltech grad to accused gunman in D.C. attack Cole Tomas Allen selfie before the attack in Washington, D.C., according to a pretrial detention memo filed by prosecutors Wednesday. This is read by an automated voice. Please report any issues or inconsistencies here . A quiet, respected tutor and engineer from Southern California with a "godly" upbringing allegedly attempted to assassinate President Trump at the White House correspondents' dinner, shocking those who knew him. Allen's social media accounts under the handle "coldForce" show years of posts criticizing Trump and supporting Ukraine, but contain no indication of violent intent despite the alleged assassination plot.

OpenAI's Sam Altman, left, is a backer of Rain Neuromophics, one of the companies that received funds from the UK's Aria, the brainchild of Dominic Cummings, right OpenAI's Sam Altman, left, is a backer of Rain Neuromophics, one of the companies that received funds from the UK's Aria, the brainchild of Dominic Cummings, right Exclusive: Brainchild of Dominic Cummings, Aria is aimed at funding'crazy' scientific projects to benefit the UK Britain's "invention agency" has pledged £50m of UK taxpayer money to US tech companies and venture capital projects. Dreamed up by Dominic Cummings to fund "crazy" ideas, the Advanced Research and Invention Agency (Aria) is meant to " restore Britain's place as a scientific superpower ". But a joint investigation by the Guardian and Democracy for Sale, an investigative website, has established that more than an eighth of the agency's £400m in research and development funding over the past two years has gone to 14 US tech companies and venture capital groups, in some cases, with no clear return for the UK or Aria. One of these companies, Rain Neuromorphics, is also backed by the OpenAI chief executive, Sam Altman, and was reported to be near collapse last year, shortly after winning Aria money. It did not respond to a request for comment; two of its founders appear to have left the company.