We consider statistical tasks in high dimensions whose loss depends on the data only through its projection into a fixed-dimensional subspace spanned by the parameter vectors and certain ground truth vectors. This includes classifying mixture distributions with cross-entropy loss with one and two-layer networks, and learning single and multi-index models with one and two-layer networks. When the data is drawn from an isotropic Gaussian mixture distribution, it is known that the evolution of a finite family of summary statistics under stochastic gradient descent converges to an autonomous ordinary differential equation (ODE), as the dimension and sample size go to $\infty$ and the step size goes to $0$ commensurately. Our main result is that these ODE limits are universal in that this limit is the same whenever the data is drawn from mixtures of arbitrary product distributions whose first two moments match the corresponding Gaussian distribution, provided the initialization and ground truth vectors are coordinate-delocalized. We complement this by proving two corresponding non-universality results. We provide a simple example where the ODE limits are non-universal if the initialization is coordinate aligned. We also show that the stochastic differential equation limits arising as fluctuations of the summary statistics around their ODE's fixed points are not universal.

Vince Zampella, who co-created the widely-popular video game Call of Duty, has died in a single-vehicle Ferrari crash in California, aged 55. Zampella's death was confirmed by Electronic Arts, which owns Respawn Entertainment, a game studio he co-founded. This is an unimaginable loss, and our hearts are with Vince's family, his loved ones, and all those touched by his work, a spokesperson for Electronic Arts told the BBC. Officials said the person on the vehicle's passenger seat was ejected while the driver remained trapped. It is unclear if Zampella was driving the car.

The critically endangered chicks, Oscar and Duffy, were born at a New Jersey aquarium. Breakthroughs, discoveries, and DIY tips sent every weekday. An aquarium in New Jersey welcomed two new residents, just in time for the holidays. On December 20, staff at Adventure Aquarium in Camden revealed the recent births of Duffy and Oscar, a pair of African penguins () and some much needed good news in light of ongoing conservation concerns . "These milestones are incredibly important for the critically endangered African penguin population, and we couldn't be more proud to play a role in their future," the aquarium just outside of Philadelphia, Pennsylvania wrote in a social media post .

Russia has intensified its strikes on the southern Ukrainian region of Odesa, causing widespread power cuts and threatening the region's maritime infrastructure. Ukrainian Deputy Prime Minister Oleksiy Kuleba said Moscow was carrying out systematic attacks on the region. Last week, he warned that the focus of the war may have shifted towards Odesa. President Volodymyr Zelensky said the repeated attacks were an attempt by Moscow to block Ukraine's access to maritime logistics. Earlier in December, Russian President Vladimir Putin threatened to sever Ukraine's access to the sea as retaliation for drone attacks on tankers of Russia's shadow fleet in the Black Sea.

Chinese robotaxis could be set to hit UK roads in 2026 as ride-sharing apps Uber and Lyft announce partnerships with Baidu to trial the tech. The two companies are hoping to obtain approval from regulators to test the autonomous vehicles in London. Baidu's Apollo Go driverless taxi service already operates in dozens of cities, mostly in China, and has accrued millions of rides without a human behind the wheel. Transport secretary Heidi Alexander said the news was another vote of confidence in our plans for self-driving vehicles - but many remain sceptical about their safety. We're planning for self-driving cars to carry passengers for the first time from spring, under our pilot scheme - harnessing this technology safely and responsibly to transform travel, Ms Alexander said in a post on X .

President Donald Trump speaks in the Roosevelt Room flanked by Masayoshi Son, Larry Ellison, and Sam Altman at the White House on January 21, 2025. President Donald Trump speaks in the Roosevelt Room flanked by Masayoshi Son, Larry Ellison, and Sam Altman at the White House on January 21, 2025. In case you missed it, 2025 was a big year for AI. It became an economic force, propping up the stock market, and a geopolitical pawn, redrawing the frontlines of Great Power competition. It had both global and deeply personal effects, changing the ways that we think, write, and relate.

Long the province of the ultra-wealthy, prenuptial agreements are being embraced by young people--including many who don't have all that much to divvy up. More than forty per cent of millennials and Gen Z-ers claim to have signed a prenup. Andrea Zevallos declared 2016 her "year of dating." She was twenty-seven, working at Universal Studios Hollywood, the theme park, and determined to find love. She calculated it would take three dates a week. By December, she was losing hope. "It was exhausting," she said. Then, while scrolling OkCupid, she noticed a "cute guy" with a "Hamilton" reference in his handle. His name was Alex Switzky, and like her he was a musical-theatre enthusiast and aspiring screenwriter. He was different from the other men she'd met. On their second date, he started planning a third. Zevallos "was used to L.A. guys cagey about any sort of calendar." One day, Switzky called her. Accustomed to texts, she assumed that he was about to break up with her. "The most millennial response," she recalled, laughing.

Devastating truth about Rob Reiner's daughter Romy: Her own addiction battle... how she'lived in fear' of Nick... and the handsome companion she's leaning on, all revealed by heartbroken friends Baby-faced accused killers will be tried as adults after 14-year-old girl's horrific murder Trans killer, 30, who executed her parents then converted to Islam is jailed for 25 years after trying to skip'stressful' sentencing I'm Miley Cyrus's REAL mother: Woman at center of bombshell'adoption' lawsuit breaks silence about'pregnancy at age 12' and makes MORE wild claims School bus driver responds to backlash after she was fired over'English-only' sign Six common medications you should NEVER mix with alcohol: Doctors reveal how that'pre-emptive' painkiller could destroy your liver... and the most deadly combination of all Next domino falls in Michigan's Sherrone Moore scandal as top assistant defects to SEC school The extravagant gifts the rich are buying this Christmas including an'extra person' in their marriage I was forced into Witness Protection at age seven... here's how the program nearly ruined my life Former Nickelodeon star is now'homeless on the streets of Los Angeles' How Tom Brady REALLY feels about Gisele Bundchen's secret wedding to jiu-jitsu instructor... as insiders whisper about potential of his OWN second marriage The hidden blueprint to keep MAGA in power for 100 years as Trump's inner circle shows signs of cracking Kimberly Guilfoyle's'yelling fit' after ex Donald Trump Jr's new engagement... as insiders reveal her nasty texts and derogatory nickname for Bettina Anderson Smug because you love your job? READ MORE: Scientists reveal surprising secret behind Bill Gates' success The saying goes, if you find a job you love you'll never work a day in your life. But an expert has now warned that this can backfire - and the seemingly innocent idea of loving your work can take on a moral edge. Mijeong Kwon, assistant professor of management at Rice University in Texas, said the dream of enjoying your career has become compulsive for many. 'Working for money, prestige or family obligation starts to look less admirable, even suspect,' she wrote on The Conversation .

US and Ukrainian envoys say productive and constructive talks have taken place in Miami, but there still appears to be no major breakthrough in efforts to end Ukraine's war with Russia. Donald Trump's special envoy, Steve Witkoff, issued a joint statement with the top Ukrainian negotiator, Rustem Umerov, after three days of meetings with European allies. The pair said the meeting focused on aligning positions on a 20-point plan, a multilateral security guarantee framework, a US Security guarantee framework for Ukraine and an economic & prosperity plan. Separate talks have been taking place in Miami between the US and the Russian envoy, Kirill Dmitriev. Our shared priority is to stop the killing, ensure guaranteed security, and create conditions for Ukraine's recovery, stability, and long-term prosperity, Witkoff and Umerov said in a statement.

The team even extracted DNA from a tiny 100-year-old butterfly leg. Breakthroughs, discoveries, and DIY tips sent every weekday. When you think of butterflies, chances are you imagine unmistakable insects with bright, bold wings. But it turns out that individual butterfly species are sometimes shockingly difficult to tell apart. "Thanks to the genetic revolution and the collaboration of researchers and museums in various countries led by London's Natural History Museum, century-old butterflies are now speaking to us," Christophe Faynel, an entomologist at the Société entomologique Antilles Guyane, said in a statement .