This paper proposes a general machine learning framework called the localization method, which is fundamentally built on two core concepts: localization kernels and local means -- key components that underpin the self-attention mechanism. To establish a rigorous theoretical foundation, the framework is formally defined through two essential pillars: the formulation of the local(-ized) model and the localization trick. We systematically investigate the connections between the localization method and a wide range of existing machine learning models/methods, including (but not limited to) kernel methods, lazy learning, the MeanShift algorithm, relaxation labeling, Hopfield networks, local linear embedding (LLE), fuzzy inference, and denoising autoencoders (DAEs). By dissecting these relationships, we clarify the broader theoretical significance of the localization method and demonstrate its practical applicability across diverse machine learning tasks. Furthermore, we explore advanced extensions of the framework, such as adaptive kernels, hierarchical local models, and non-local models. Notably, we show that the Transformer -- a cornerstone of modern sequence modeling -- can be constructed using hierarchical local models, revealing the ability of the localization method to unify and generalize state-of-the-art architectures. This work not only provides a unified theoretical lens to reinterpret existing models but also offers new methodological tools for designing flexible, data-adaptive learning systems.

We analysed thousands of Trump's posts - here's what we found In 2026, Donald Trump's use of social media has escalated. The BBC sifted through thousands of posts on his platform Truth Social to analyse what the President has been saying and when. What was the busiest day? When are the busiest hours? What type of content does President Trump share?

Paul McCartney on playing guitar with Paul Mescal: 'He knew it better than I did!' Hey, I know you! exclaims Paul McCartney, gripping my hand as we walk into his office in central London. And while I'm realistic enough to know he doesn't really hold treasured memories of our previous encounters, I'm impressed by his ability to defuse the tension of Meeting A Beatle. We gather in Soho at lunchtime. Instead of Wild Honey Pie or Savoy Truffle, McCartney has opted for a simple bagel (topping: a terrifying blend of Marmite and hummus), which he prepared in a kitchenette next to his assistant's desk. As he eats, he scans a printed list of film titles - mainly vintage comedies - looking for something to play at his family movie night.

According to federal prosecutors, Michele Spagnuolo made more than $1 million on the prediction market platform using confidential information about Google Search traffic. A Google security engineer has been charged with crimes stemming from allegedly placing trades on Polymarket using confidential internal information from the tech giant. Michele Spagnuolo, a 36-year-old Italian citizen, was arrested this morning in New York, as first reported by ABC News. Spagnuolo is charged with one count each of commodities fraud, wire fraud, and money laundering. He has worked at Google since 2014 and was based out of the company's Zurich, Switzerland, offices.

The world's carmakers are struggling to compete with China Global carmakers are facing a reckoning as US, European and Japanese brands lose ground to Chinese rivals setting the pace not only in electric vehicles, but also in batteries, design and software. The BBC visited factory floors in Beijing and Hefei on the sidelines of Auto China 2026 - the world's largest car show - and found striking levels of automation and software development speed, leaving foreign brands that once dominated the Chinese market struggling to keep up. We have no chance against this, Honda chief executive Toshihiro Mibe told Japanese media after visiting a highly automated factory in Shanghai. Ford chief executive Jim Farley has also warned that Western carmakers, are in a fight for our lives as Chinese rivals expand globally. After decades spent investing in joint ventures with Chinese partners to build vehicles, foreign carmakers are now changing the nature of those partnerships to stay competitive.

One in six young people will not be in education, employment or training within five years unless urgent action is taken, a major review has warned. The education, health and welfare systems are no longer fit for purpose in preparing young people for adult life, said its author former minister Alan Milburn. We are at risk of a lost generation, he warned, with the number of 16 to 24-year-olds out of work, education or training set to rise to 1.25 million by 2031. The first rung of the career ladder has thinned and that for too many young people it is now simply out of reach, Milburn is set to say in a speech later. That places them in a hopeless catch-22 where employers ask for work experience but the opportunities for young people to gain it have narrowed or gone, he will say.

NASA's MoonFall mission plans to send drones to the moon's South Pole sometime in 2028. More information Adding us as a Preferred Source in Google by using this link indicates that you would like to see more of our content in Google News results. Breakthroughs, discoveries, and DIY tips sent six days a week. By signing up, you confirm you are 16+, will receive newsletters and promotional content and agree to our Terms of Use and acknowledge the data practices in our Privacy Policy . As NASA looks ahead towards Artemis III in mid-2027, the agency is sharing new details on several projects, including a future permanent moon base and a drone mission called MoonFall.

Russian lawmakers have passed a bill to allow trained bank employees to shoot down Ukrainian drones amid an increase in the number of attacks. The draft legislation, which would see banks across Russia install electronic jamming systems while selected employees would shoot down incoming unmanned aircraft, passed in its third and final reading in the lower house Duma on Tuesday, according to the state-run TASS news agency. The bill says the legislation is needed to protect Bank of Russia facilities, including those located in the new constituent entities of the Russian Federation - referring to the four eastern Ukrainian regions that Moscow has announced it has annexed despite not controlling them fully - amid the increasing number of sabotage and terrorist attacks. Under the plan, banks would finance the installation of the equipment on their premises. With banks in almost every town, their incorporation into Russia's air defences could help expand its cover.

Belgium and France have summoned Russia's ambassadors to express anger after Moscow urged foreigners to leave Kyiv in advance of planned "systematic strikes". Brussels and Paris said on Wednesday that Russia's announcement earlier in the week was "unacceptable" and a violation of international law. The spat is unlikely to help smooth the way for the EU to mediate talks to bring the conflict to an end, an arrangement that Russian President Vladimir Putin said on Wednesday he is ready to accept. Germany, the Netherlands, Norway and the European Union summoned Russian envoys on Tuesday following Moscow's warning that foreigners and diplomats should leave the Ukrainian capital before the onset of renewed air strikes . "Threatening embassies is not diplomacy, it is intimidation. And it is a flagrant violation of international law and the Vienna Convention," Belgium Foreign Minister Maxim Prevot said on Wednesday.

PCWorld reports the Steam Deck has returned to market with nearly doubled pricing, featuring only OLED models at $789 for 512GB and $949 for 1TB versions. Valve discontinued cheaper LCD models and attributes price increases to rising memory and storage costs driven by AI industry demand affecting consumer hardware. This trend extends beyond Steam Deck to other gaming handhelds like Lenovo Legion Go, signaling broader affordability challenges in PC gaming hardware. The Steam Deck, harbinger of a portable PC gaming revolution, has been out of stock for three months. Now it's back at almost double the price of the original model. It'll cost you $789 USD to get the 512GB OLED version, $949 for the 1TB upgrade. The original LCD model, which debuted at $400, is resigned to the dustbin of history . Welcome to PC gaming in 2026, where the K-shaped economy has claimed the last remaining affordable option. The 512GB OLED model now costs $1,129 in Canada, 649 pounds in the UK, 779 euro in Europe, $1,199 in Australia, and 3,279 PLN in Poland.