Variational inference (VI) is a central tool in modern machine learning, used to approximate an intractable target density by optimising over a tractable family of distributions. As the variational family cannot typically represent the target exactly, guarantees on the quality of the resulting approximation are crucial for understanding which of its properties VI can faithfully capture. Recent work has identified instances in which symmetries of the target and the variational family enable the recovery of certain statistics, even under model misspecification. However, these guarantees are inherently problem-specific and offer little insight into the fundamental mechanism by which symmetry forces statistic recovery. In this paper, we overcome this limitation by developing a general theory of symmetry-induced statistic recovery in variational inference. First, we characterise when variational minimisers inherit the symmetries of the target and establish conditions under which these pin down identifiable statistics. Second, we unify existing results by showing that previously known statistic recovery guarantees in location-scale families arise as special cases of our theory. Third, we apply our framework to distributions on the sphere to obtain novel guarantees for directional statistics in von Mises-Fisher families. Together, these results provide a modular blueprint for deriving new recovery guarantees for VI in a broad range of symmetry settings.

An autonomous robot from the company Honor ran a half marathon in 50:26, beating the human record by 7 minutes. A humanoid robot from the Honor remote-controlled team crosses the finish line during the E-Town Humanoid Robot Half Marathon in Beijing on April 19, 2026. Over the weekend in China, a humanoid robot shattered world half-marathon record--the human record--by seven minutes. The star performer was a robot developed by the Chinese company Honor (the smartphone maker), which finished the 13.1-mile race in 50 minutes, 26 seconds. The human record, set by Ugandan Olympic medalist Jacob Kiplimo, is 57 minutes, 20 seconds.

Outside of Japan, Google's chatbot is accessible on both desktop and iOS. Google has added a new sidebar to Chrome that allows users to access Gemini from any of their tabs. After debuting in the US, Gemini in Chrome is making its way to more markets. Starting today, Google is rolling out Chrome's built-in chatbot to users in Asia and the Pacific, including Australia, Indonesia, Japan, the Philippines, Singapore, South Korea and Vietnam. The expansion comes after Google earlier this year made Gemini in Chrome available to people in Canada, India and New Zealand .

John Ternus, the company's senior vice president of hardware engineering, will replace Cook as CEO on September 1. Cook will stay on as executive chairman. Tim Cook is stepping down as the CEO of Apple and transitioning to a role as the company's executive chairman, effective September 1, the company announced on Monday. John Ternus, Apple's senior vice president of hardware engineering, will replace Cook as CEO . Cook's departure had been speculated upon in recent months. In an era when every other Big Tech company has thrown significant resources at developing advanced AI, Apple is widely perceived as a laggard.

MathNet contains 30,000 free math problems collected over half a century. 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. The International Mathematical Olympiad was first held in Romania in 1959. Breakthroughs, discoveries, and DIY tips sent six days a week. In 1959, countries around the world sent their most talented students to Romania to compete in the first-ever International Mathematical Olympiad (IMO).

The actor-director discussed his least favorite currency and read a series of mean tweets--about us!--at our inaugural WIRED@Night event. Ben McKenzie had a question: "When did WIRED die?" Specifically, the actor-director wanted to know when did WIRED "'DIE,' all caps." McKenzie wasn't asking for himself; he was engaging in the time-honored celebrity tradition of reading mean tweets . McKenzie, who famously played Ryan on before becoming a leading voice of crypto skepticism, was sharing the stage with WIRED senior correspondent Andy Greenberg for the first of what will hopefully be a series of smaller events that we are calling WIRED@Night. On April 16, about 100 people gathered at event partner Ace Hotel Brooklyn to sip drinks from Aplos, Faccia Brutto, The Sorting Table, and Manojo and ponder the future of cryptocurrency.

Nancy Guthrie sheriff's appalling past revealed: Beat handcuffed suspect so badly he needed intensive care, used VILE language about woman and lied in sworn statement Vance grounded at White House as Iran peace talks in turmoil and Trump declares: 'I expect to be bombing' New'Hollywood dose' pill: A-listers hooked on'youth elixir' that dermatologists say is anti-ageing, shrinks pores, smooths wrinkles... and even banishes rosacea Days after we got engaged, the love of my life told me he'd killed a man and buried him in a bog. I reported him to police... but then I made this irreversible mistake Ark of the Covenant's final resting place pinpointed by archaeologists as fresh search begins Ritzy Bay Area town torn apart after teacher's daughter, 16, crashed car while speeding and killed four friends... then posted a TikTok video that poured fuel on the flames Jordon Hudson extends her control over Bill Belichick's empire with secret move that is set to leave his family and friends furious Two CIA officers killed in Mexico when their car skidded off ravine and exploded after meeting about bust of'largest ever drug lab' Life-threatening cantaloupe recall in four states upgraded to FDA's highest risk level... 'reasonable probability of death' AMANDA PLATELL: Why Sarah Ferguson - with the ghost of Princess Diana at her side - is ready to sensationally blow up the Royal Family. She knows ALL their secrets... Trump confronts Xi as US forces seize Chinese ship carrying mysterious'gift' to Iran Team USA Olympics star Noah Lyles slammed for'horrible' reaction to his wife's wedding dress reveal In honour of the Queen's (purple!) reign: Kate mirrors late monarch's colourful wardrobe and wears her pearl earrings and necklace How to lose weight when perimenopause sabotages your metabolism: I'm a trainer but when I hit 46, I piled on the pounds overnight. The new'posh' drug that's easier to order than Uber Eats - and why all my middle-class friends have ditched booze and cocaine for it: JANA HOCKING Fury as murderer marries pen pal behind bars... as teenage victim's mom says: 'I'm serving a life sentence without my son' Autistic woman, 24, worked hard to build independent life for herself... now she's PARALYZED thanks to selfishness of stranger Chilling warning from Nobel physicist as date is set for humanity's final destruction The winner of the Nobel Prize in Physics has issued a stark warning to humanity, saying it could face an existential catastrophe within roughly 35 years. David Gross, who shared the 2004 Nobel Prize in Physics, warned that'due to the danger of nuclear war,' humankind may have just a little more than three decades left.

The systems behind them--and the reasons we keep passing around war memes as entertainment--are more serious. As ceasefire announcements between the US and Iran --and separately between Israel and Lebanon --dominated headlines over the past two weeks, they also prompted a look back at how war spread online: through memes. There were jokes about conscription. Captions about getting drafted, but at least with a Bluetooth device. The song "Bazooka" went viral, with users lip-syncing to "Rest in peace my granny, she got hit by a bazooka."

Elon Musk was summoned by the French authorities for a voluntary interview that had been planned to take place on Monday. Elon Musk was summoned by the French authorities for a voluntary interview that had been planned to take place on Monday. Elon Musk did not appear on Monday for a voluntary interview with lawyers in Paris, who had summoned the American tech billionaire over an investigation into his social media platform X and AI chatbot Grok. The prosecutors told AFP that they had "taken note of the absence of the first people summoned", without mentioning Musk's name. The billionaire called the French authorities involved "retards" weeks earlier in a French-language post on X .

Get your news from a source that's not owned and controlled by oligarchs. On Sunday afternoon, Palantir, the defense-tech company that sells software to clients like ICE, the US military, and the Israeli military, decided to give us all a piece of their mind. The company's official X account published a list of excerpts from co-founder Alex Karp's 2025 book The book frames Silicon Valley's move into military technology as the righteous repayment of a "moral debt" owed to the country that built the tech billionaire class. "The engineering elite of Silicon Valley has an affirmative obligation to participate in the defense of the nation." If you read past the post and dig into the book itself, you'll find that this sentence continues: "the engineering elite must also, Karp said, participate in "the articulation of a national project--what is this country, what are our values, and for what do we stand." That is to say: Men like Karp should decide what this country is. "If a US Marine asks for a better rifle, we should build it; and the same goes for software," Palantir's Bill-Ackman-esque digression continued. It asserts that the future of American military dominance will not depend on nuclear deterrence, but on AI weaponry--possibly like the Palantir AI product that is reportedly used to help generate'kill lists' for the Israeli military in Gaza. Then, after arguing for the primacy of its own products--called " spy tech " by Palantir's critics--Karp suggests the remilitarization of the Axis Powers. "The postwar neutering of Germany and Japan must be undone," Karp's company account asserted. "The defanging of Germany was an overcorrection for which Europe is now paying a heavy price.