Trump's hidden five-tap code in handshake with Xi... and the tell-tale'bullfrog smile' that betrayed the president How Kylie Kelce REALLY feels about Taylor Swift after her foul-mouthed wedding rant: Insiders reveal stark'differences' between the sisters-in-law... and truth about'feud' What REALLY goes on in some Equinox steam rooms: Gym insiders reveal eye-popping indecency... secret towel signals used by experimental married men... and clubs with most'aggressive' locker rooms Xi greets Trump with ominous warning about risk of war between US and China: 'Beware the Thucydides Trap' Inside Eric Swalwell's marriage implosion: Disgraced Democrat nowhere in sight at family home as his furious wife appears without her ring and delivers ultimate insult Home Depot and Lowe's use sneaky cameras in theft crackdown - but honest customers are the real victims Buster Murdaugh's explosive reaction as his father's murder conviction is overturned: Insiders reveal all about his secret new life... and jailhouse calls with Alex Inside Carrie Underwood's'grounded' and'traditional' home life on her 400-acre Tennessee farm Grotesque new Michael Jackson allegations raise questions about his accusers so taboo they're almost impossible to ask... but we must: MAUREEN CALLAHAN Kylie Jenner and Timothee Chalamet's mortifying relationship secrets exposed: Her'jealousy'... his pleas for'space'... and why he's now finally'on board' with a proposal Beautiful young mom appeared to have it all. Now her two toddlers are dead after falling into a pool while on COCAINE... and her own parents allegedly made very troubling comments about her Walmart axes 1,000 workers as white-collar jobs bloodbath reaches America's biggest private employer Grief author Kouri Richins gives 40-minute rant about love and calls husband's poisoning murder a'tragedy' as she learns fate in Moscow Mule slaying... and sends deranged message to her sons This miracle drug rapidly reversed my balding. It wrecked my sex life... but a microdosing hack gave me my libido and my hair back Huge asteroid with the potential to obliterate an entire CITY will narrowly miss Earth - as scientists warn'it's as close as you can get without hitting' An asteroid with the potential to obliterate a city will narrowly miss Earth next week, scientists have revealed. Astronomers say the space rock, called 2026 JH2, is up to four times the size of a London bus and will get'as close as you can without hitting'. It is expected to zoom by our planet at an estimated distance of around 56,000 miles (90,000km) late on Monday night.

'Our characters are cybernetic or larger than life,' said Zak London, the founder of Gossip Goblin. 'We adapt to the limits of AI acting.' 'Our characters are cybernetic or larger than life,' said Zak London, the founder of Gossip Goblin. 'We adapt to the limits of AI acting.' 'There are no rules': spotlight on Gossip Goblin as AI film-making enters new era Defying criticisms of'slop' and'theft', the growing culture of AI-powered creativity is attracting interest from Hollywood In a former hemstitching workshop where artisans sewed pleats for Stockholm's 19th-century bourgeoisie, a distinctly 21st-century craft is taking root: AI film-making. One day last week, an actor, director and composer squeezed into a tiny studio booth to record a voiceover for their next AI release. But this had a distinctly homespun feel, the little team fussing over a monologue by a poetic Scottish gorilla inhabiting a transhumanist cyberpunk universe.

One person has been killed and dozens wounded after Russia launched a massive wave of strikes against Ukraine overnight, officials have said. Prime Minister Yulia Svyrydenko said it was an especially difficult night for Kyiv, where rescuers are searching through the rubble of a residential building after a combined missile and drone attack on the Ukrainian capital. The overnight barrage followed a major attack on Wednesday - one of the largest Russia has mounted since the start of its full-scale invasion in 2022, with 892 drones launched in 24 hours. It marks the third day in a row Ukraine has reported deaths, as Moscow ramps up its assault after a three-day ceasefire expired on Monday. In the capital, a search and rescue operation began early Thursday to look for people under the rubble of a nine-storey apartment block which had been hit overnight.

Apple is on Google's side when it comes the latter's criticisms against the European Union's proposals which would give third-party AI services the same level of access to Android that Gemini has. The European Commission has been taking steps to ensure that Google complies with the rules of the Digital Markets Act (DMA). In January, it told the company that it has to give external AI assistants the same access to Android its own technology has and to hand over anonymized ranking, query, click and view data held by Google Search to rival search engines. The aim, the commission said, was to give third-party providers an an equal opportunity to innovate and compete in the rapidly evolving AI landscape on smart mobile devices. It explained that opening up Android would keep the AI market open and promote innovation in the field.

'I was given a choice - keep my legs or keep my life' - the sepsis patient who lived Farmer Marshall Wylie thought nothing of it when he cut his arm, sorting wood in August 2023. And he thought even less of it when he felt ill over the next 48 hours. But the following week, he said he clinically died due to sepsis, and eventually his legs had to be amputated. Farmers are at particular risk of developing sepsis due to incidents on the farm, but can also be reluctant to seek healthcare. Warning: This article contains some graphic images of hands and feet with sepsis.

HM Revenue and Customs has announced a 10-year, £175m deal with the British tech firm Quantexa to provide AI-powered technology to help improve its performance. Quantexa says its systems will combine data collected by HMRC with external sources to help the tax office identify incidents of fraud and fix unintentional errors more quickly. Its tasks will include helping HMRC to assist customer service staff, as well as to identify hidden networks of companies and individuals masking fraudulent activity. Public dissatisfaction with HMRC performance has crept up in recent years, according to government figures. A Freedom of Information request made by the campaigners at the Contentious Tax Group found there were more than 93,000 complaints made about the department in 2024-2025 .

Trump's hidden five-tap code in handshake with Xi... and the tell-tale'bullfrog smile' that betrayed the president How Kylie Kelce REALLY feels about Taylor Swift after her foul-mouthed wedding rant: Insiders reveal stark'differences' between the sisters-in-law... and truth about'feud' What REALLY goes on in some Equinox steam rooms: Gym insiders reveal eye-popping indecency... secret towel signals used by experimental married men... and clubs with most'aggressive' locker rooms Xi greets Trump with ominous warning about risk of war between US and China: 'Beware the Thucydides Trap' Inside Eric Swalwell's marriage implosion: Disgraced Democrat nowhere in sight at family home as his furious wife appears without her ring and delivers ultimate insult Home Depot and Lowe's use sneaky cameras in theft crackdown - but honest customers are the real victims Buster Murdaugh's explosive reaction as his father's murder conviction is overturned: Insiders reveal all about his secret new life... and jailhouse calls with Alex Inside Carrie Underwood's'grounded' and'traditional' home life on her 400-acre Tennessee farm Grotesque new Michael Jackson allegations raise questions about his accusers so taboo they're almost impossible to ask... but we must: MAUREEN CALLAHAN Kylie Jenner and Timothee Chalamet's mortifying relationship secrets exposed: Her'jealousy'... his pleas for'space'... and why he's now finally'on board' with a proposal Beautiful young mom appeared to have it all. Now her two toddlers are dead after falling into a pool while on COCAINE... and her own parents allegedly made very troubling comments about her Walmart axes 1,000 workers as white-collar jobs bloodbath reaches America's biggest private employer Grief author Kouri Richins gives 40-minute rant about love and calls husband's poisoning murder a'tragedy' as she learns fate in Moscow Mule slaying... and sends deranged message to her sons This miracle drug rapidly reversed my balding. It wrecked my sex life... but a microdosing hack gave me my libido and my hair back MORE: FBI files reveal reports of'four-foot tall' beings emerging from UFOs Scientists have announced a groundbreaking discovery lurking beyond our solar system that rewrites humanity's understanding of the cosmos. A new study led by Princeton University in New Jersey identified more than 10,000 new possible planets trillions of miles from Earth. That included at least 11 worlds described by scientists as'super-Earths.'

We establish the first population risk bounds for Kolmogorov-Arnold Networks (KANs) trained by mini-batch SGD with gradient clipping, covering non-private SGD as well as differentially private SGD (DP-SGD) with Gaussian perturbations that interpolate between independent and temporally correlated noise. This setting is substantially closer to practice than prior KAN theory along two axes: training is by mini-batch SGD, the standard recipe for modern networks, rather than full-batch gradient descent (GD); and correlated-noise mechanisms have empirically shown a more favorable privacy-utility tradeoff than independent-noise mechanisms. Our results cover the corresponding full-batch GD and independent-noise DP-GD results for KANs by Wang et al. (2026), while yielding sharper fixed-second-layer specializations. The technical core is a new analysis route for correlated-noise DP training in the non-convex regime. Temporal dependence breaks the conditional-centering structure underlying standard one-step SGD arguments, and the projection step obstructs the exact cancellation structure of correlated perturbations. We address these difficulties through an auxiliary unprojected dynamics, a shifted iterate that absorbs the current noise perturbation, and a high-probability bootstrap certifying projection inactivity. Combining this optimization analysis with a stability-based generalization argument yields the stated population risk bounds. To the best of our knowledge, this is the first optimization and population risk analysis of a correlated-noise mechanism for DP training beyond convex learning, in particular for neural networks.

While deep learning has revolutionised inverse problems, its safe deployment is hindered by three primary reliability concerns: hallucinations, instabilities, and performance volatility [48]. Hallucinations manifest as high-fidelity features that are factually false; instabilities reflect heightened sensitivity to measurement noise; and performance volatility refers to significant fluctuations in reconstruction quality across the data, yielding high-fidelity results for some samples while failing on seemingly similar images. In many applications, the risk of generating realistic but unfaithful content can impede the safe deployment of AI methods for inverse problems. The choice of "hallucinate" as the Cambridge Dictionary's word of the year in 2023 illustrates this open problem [53]. The problem of AI hallucinations persists, as the Financial Times [44] highlighted that, "AI hallucinations haunt users more than job losses." A first step toward training AI methods that do not suffer from hallucinations is the assessment and identification of hallucinated outputs. Consider the inverse problem of recovering xfrom noisy measurements y " Fpx,eq, x PM1 ĂX, e PEĂY, (1.1)

Discrete automated processes in industrial and cyber-physical systems often exhibit a repetitive structure in which successive repetitions follow a common trajectory while differing in duration, amplitude, and fine-scale dynamics. Such \emph{approximately periodic} behavior poses a challenge for Gaussian Processes (GP) modeling: strictly periodic models suppress inter-repetition variability, while non-periodic models fail to capture the strong structural regularities required for generation. In this work, we propose a stochastic generative model for approximately periodic time series. The model is based on a GP whose posterior is modulated by a novel kernel. Our approach decouples intra-repetition structure from inter-repetition variability through a two-stage construction which yields a generative distribution with a identical mean function across repetitions, while allowing smooth variation between repetitions. The modeling choices are supported by an implementation in which realistic synthetic trajectories are generated from toy datasets.