We study the sublinear multivariate mean estimation problem in d-dimensional Euclidean space. Specifically, we aim to find the mean µ of a ground point set A, which minimizes the sum of squared Euclidean distances of the points in Ato µ. We first show that a multiplicative (1 + ε) approximation to µ can be found with probability 1 δ using O(ε 1 logδ 1)many independent uniform random samples, and provide a matching lower bound. Furthermore, we give two estimators with optimal sample complexity that can be computed in optimal running time for extracting a suitable approximate mean: 1.

Former Olympian seen in handcuffs as Trump threatens'years in jail' and more arrests after vandals SABOTAGE Reflecting Pool with'corrosive and destructive chemicals' Keir Starmer'will announce as early as Monday that he is quitting as Prime Minister' after spending weekend locked in tense talks about his future with his wife Victoria at Chequers Mortifying truth about Clavicular's'botched' nose job: Infertile influencer's'trans' admission to friends... as insider reveals what's said behind closed doors - and twisted secrets that'll leave fans floored Giorgia Meloni rips'senseless' attacks from Trump as Italian Prime Minister refuses to back down amid G7 feud Inside America's new fattest town: Burgers are the size of your head, gyms lie empty and custom mobility scooters carry 800lb loads... as we investigate why Ozempic just DOESN'T work Call me cynical, but the real reason Gruesome Twosome Harry and Meghan are returning to the UK is just so obvious... and highly humiliating: MAUREEN CALLAHAN Stingy fast food giant named America's favorite restaurant AGAIN... and experts think they know why I lost 50lb without jabs using this easy but overlooked method. But I still felt dowdy - until I discovered these expert anti-ageing fashion and beauty tips. Embattled Alexi Lalas makes controversial World Cup declaration amid tension with Fox colleagues: 'Makes you look like a weak poser' No one can see the real reason Jelly Roll divorced Bunnie XO. Angelina Jolie's son Pax, 22, surfaces in LA after bombshell revelation about his relationship to Brad Pitt Scientists propose radical new theory of consciousness - and claim it doesn't depend on flesh and blood Candace Owens hits out at nasty rumors claiming she was DEAD... as fellow MAGA influencer claims her account was hacked Blake Lively runs errands in frumpy outfit after reconciling with ex-BFF Taylor Swift... miles away from reported'bachelorette party' Grace Kelly's lookalike granddaughter, 27, wows in bikini snaps...as she packs on the PDA during beach getaway TV star mom, 46, who appeared on'quitting everything to change your life' show died in fire at luxury Caribbean beach resort that sent 1,700 tourists running for their lives The four mistakes that led to bungee tragedy on Skeleton Bridge: FRED KELLY saw the scene for himself, now he retraces the prelude to disaster. So was it really an accident?

The arachnids have multiple layers of retinas in each eye. 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 . Technological advancements sometimes feel like a perpetual fight against nature itself, with scientists working out how to do things faster, more powerfully, and more efficiently.

Plus: The EU has proposed new legislation to end its Big Tech dependence. Most days in her chambers, Judge Maritza Braswell, a federal magistrate judge in Colorado, sifts through stacks of documents written by people without a lawyer. The number of these filings has more than doubled compared to before 2023. She puts that jump down to AI. But while AI appears to be expanding access to justice, it doesn't seem to be improving people's chances of winning. Judges are starting to question what rights and duties chatbots should have as they stand in for lawyers.

The world's first'hovertrain' could reach speeds of 270 mph in the 1960s But the futuristic Aérotrain never saw the light of day. 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. This cancelled Mongolian postage stamp shows the Aérotrain Orleans, circa 1979. 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 .

The US Built a Site to Ensure Fair Access to Public Lands. Recreation.gov was supposed to make access to public lands more equitable and streamlined. It's a few minutes before 8 am Mountain Time on March 16, the day that river permit cancellations are released on Recreation.gov, the federal website for public land reservations. Rec.gov, as it's commonly called, administers everything from river permits and timed entrance fees at the most popular national parks to campground reservations on remote sites belonging to the Bureau of Land Management, and a lot of people are recreating on public land these days. There were 11 million reservations on the site in 2024, up significantly from 3.5 million reservations reported in 2019. At the center of it all is an unlikely player in the outdoor recreation space: The site is operated by the government contractor Booz Allen Hamilton, a corporation known more for cybersecurity than rafting trips. Early each year, outdoor enthusiasts gear up for Recreation.gov's annual lotteries for some of the most iconic experiences in the country: a river trip down Idaho's Middle Fork of the Salmon River, which flows through the Frank Church River of No Return Wilderness. Backcountry permits to hike into the Wave, an otherworldly rock formation in Arizona's Paria Canyon-Vermilion Cliffs Wilderness. Overnight stays in the rugged, lake-studded Enchantments, in Washington's Okanogan-Wenatchee National Forest. Odds of getting a desirable Middle Fork permit are around 2 percent.

The bioluminescent plants are a potential alternative to electrical light and batteries. 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. Acidic (top) and basic (bottom) environments trigger different bioluminescent behaviors in algae. Breakthroughs, discoveries, and DIY tips sent six days a week. Bioluminescence is everywhere in nature, but it puts on its biggest light shows underwater .

The Department of Justice has announced that it's intervening on the behalf of xAI in the company's recent lawsuit against the state of Colorado. The law is set to go into effect in June, and the DOJ is now asking a Colorado District Court to declare it unconstitutional. In xAI's original argument, Colorado Bill SB24-205 violated the company's First Amendment rights by forcing its developers to change how they create AI products and compelling them to align their products with Colorado's views on diversity and discrimination. The DOJ acknowledges those concerns in its complaint, but specifically focuses its argument on the idea that the law violates the Equal Protection Clause of the Fourteenth Amendment. According to the DOJ, because the law relies on demographics and statistical disparities as evidence of discrimination, it will essentially require developers to distort an AI system's outputs and discriminate based on race, sex, religion and other protected characteristics, a violation of the Fourteenth Amendment.

The DOJ joined Elon Musk's xAI in suing Colorado, alleging a state AI regulation law violates the First and Fourteenth amendments by forcing developers to adopt DEI ideology.