Estimating the number of components is a fundamental challenge in unsupervised learning, particularly when dealing with high-dimensional data with many components or severely imbalanced component sizes. This paper addresses this challenge for classical Gaussian mixture models. The proposed estimator is simple: center the data, compute the singular values of the centered matrix, and count those above a threshold. No iterative fitting, no likelihood calculation, and no prior knowledge of the number of components are required. We prove that, under a mild separation condition on the component centers, the estimator consistently recovers the true number of components. The result holds in high-dimensional settings where the dimension can be much larger than the sample size. It also holds when the number of components grows to the smaller of the dimension and the sample size, even under severe imbalance among component sizes. Computationally, the method is extremely fast: for example, it processes ten million samples in one hundred dimensions within one minute. Extensive experimental studies confirm its accuracy in challenging settings such as high dimensionality, many components, and severe class imbalance.

Peter Molyneux OBE is reflecting upon the future of the UK games industry in his office - and how he could soon be leaving it. The 66-year-old, who over the years has helped create iconic series such as Fable, Black & White and Theme Park, tells me Masters of Albion - his latest project as creative director of 22cans - will also be his final one. He sees it as a return to his roots - a reinvention of the god game - a genre he introduced with Populous in 1989, one where players play as a deity on high, controlling a population's inhabitants as they please. In this new iteration, players are able to build and manage settlements by day, before defending them from attacks at night, with the ability to take control of individual characters at any point. For Molyneux, once voted one of the top game creators of all time, the key idea is freedom - creating systems that respond to player curiosity rather than directing them down a fixed path.

Meta to track workers' clicks and keystrokes to train AI Meta will start tracking the way employees work, including their keystrokes and mouse clicks, to train its artificial intelligence (AI) models. The company, which owns Instagram and Facebook, told workers on Tuesday that a new tool will run on Meta's computers and internal apps, logging their activity to be used as training data for AI technology. A Meta spokesman told the BBC: If we're building agents to help people complete everyday tasks using computers, our models need real examples of how people actually use them. The data is not used for any other purpose, he said, adding that the tool has safeguards in place to protect sensitive content. But one Meta employee, who asked not to be identified, said having their smallest actions on a computer being used to train AI model as workers expect a slew of additional job cuts feels very dystopian.

Do the Dodgers get an unfair advantage with'bizarre' rule impacting Shohei Ohtani? Florida's Thomas Haugh ditches Draft to return to school amid swirling Todd Golden rumors Mamdani takes'Curse of the Mambino' on the chin as Mets' 11-game skid sets franchise record Cubs' co-owner pushes back on woke backlash Matt Shaw received for attending Charlie Kirk's memorial'Zig-Zag Theory': Houston Rockets will cover and even series vs. Los Angeles Lakers in Game 2 Stephen A. Smith says he believes Dianna Russini and Mike Vrabel are innocent -- for now Wisconsin teen's turkey hunt takes a wild turn when a bobcat lunges and grabs his arm on camera Matt Fitzpatrick is the king of Harbour Town, lame USA chants, and LIV Golf's telling announcement Dana Perino: Economic pressure on Iranian regime is'excruciating' IRGC's'extreme commanders' abused Iranian people for decades: Ret Lt Col Chuck DeVore Missing scientists probe was reportedly sparked after'UFO General' disappeared Iranian leaders say they don't negotiate'under the shadow of threats' VP Vance's Pakistan trip suspended as Trump weighs diplomacy vs force on Iran Former Virginia governor condemns Dem redistricting plan as'illegal power grab' California Dems warned of'blue Armageddon' in governor race This is how the US can'pressure' the Iranian regime: Former leading CENTCOM official Democratic Senate candidate criticized for remarks about JD Vance's family A stretcher crew stood by but there wasn't much left to carry off after the robot's wild self-destruction A humanoid robot, 'Lightning,' shattered the Beijing half-marathon world record this weekend, completing the race in just 50 minutes and 26 seconds, 13 miles faster than any human. Cyber expert Kurt Knutsson warns this massive leap in artificial intelligence, along with Tesla's Optimus robot, necessitates a universal'off button' and stronger guardrails to ensure safety and prevent future human replacement. I'm equally excited and terrified by it. On one hand, I'd love to have a robot around the house so it could fold my laundry and make me feel like George Jetson.

Trump EXTENDS Iran ceasefire again as he backs off bombing threat amid chaos among'seriously fractured' Tehran leadership Anna Kepner's stepbrother skips court appearance as prosecutors fight to put him behind bars amid rape and murder charges New'Hollywood dose' pill: A-listers hooked on'youth elixir' that dermatologists say is anti-aging, shrinks pores, smooths wrinkles... and even banishes rosacea Truth about your Mounjaro injection site: Our expert doctors reveal exactly where you should inject yourself for the best results, what to do if your weight loss has slowed down... and the areas you should NEVER jab Driver who hit and killed jogger father-of-two sues victim's estate claiming incident left him with severe PTSD World Series winner and MLB great Garret Anderson's cause of death revealed after his sudden passing at 53 Sydney Sweeney's role is cut from The Devil Wears Prada 2 Alarm over popular new coffee chain invading the US... as experts warn of chilling secret behind its $1.99 brew 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 Wealthy realtor, 86, who'loved the finer things' disappeared into California desert after fight with daughter and grandson... then a livestreamer made horrific discovery at beauty spot Life-threatening cantaloupe recall in four states upgraded to FDA's highest risk level... 'reasonable probability of death' MORE: Death of Air Force whistleblower set to reveal UFO secrets declared'suspicious' One of the leading voices pushing for UFO disclosure has made a shocking connection between an ancient biblical text and the existence of alien life. Congresswoman Anna Paulina Luna of Florida recently posted two cryptic messages on X, one telling people to'Read the book of Enoch' and the other displaying the 15th-century painting nicknamed the ' Madonna of the UFO.' It is the latest reference the chairwoman of the House Oversight Committee's hearings on UFOs has made to the Book of Enoch while speaking about extraterrestrials and alien spacecraft. The book is an ancient Jewish religious text, written in stages between 300 and 100 BC, attributed to the biblical figure Enoch, the great-grandfather of Noah.

In this paper we present a practical solution with performance guarantees to the problem of dimensionality reduction for very large scale sparse matrices. We show applications of our approach to computing the Principle Component Analysis (PCA) of any n dmatrix, using one pass over the stream of its rows. Our solution uses coresets: a scaled subset of the n rows that approximates their sum of squared distances to every k-dimensional affine subspace. An open theoretical problem has been to compute such a coreset that is independent of both n and d. An open practical problem has been to compute a non-trivial approximation to the PCA of very large but sparse databases such as the Wikipedia document-term matrix in a reasonable time. We answer both of these questions affirmatively. Our main technical result is a new framework for deterministic coreset constructions based on a reduction to the problem of counting items in a stream.

Neural Information Processing Systems http://nips.cc/

Trump EXTENDS Iran ceasefire again as he backs off bombing threat amid chaos among'seriously fractured' Tehran leadership Anna Kepner's stepbrother skips court appearance as prosecutors fight to put him behind bars amid rape and murder charges New'Hollywood dose' pill: A-listers hooked on'youth elixir' that dermatologists say is anti-aging, shrinks pores, smooths wrinkles... and even banishes rosacea Truth about your Mounjaro injection site: Our expert doctors reveal exactly where you should inject yourself for the best results, what to do if your weight loss has slowed down... and the areas you should NEVER jab Driver who hit and killed jogger father-of-two sues victim's estate claiming incident left him with severe PTSD World Series winner and MLB great Garret Anderson's cause of death revealed after his sudden passing at 53 Sydney Sweeney's role is cut from The Devil Wears Prada 2 Alarm over popular new coffee chain invading the US... as experts warn of chilling secret behind its $1.99 brew 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 Wealthy realtor, 86, who'loved the finer things' disappeared into California desert after fight with daughter and grandson... then a livestreamer made horrific discovery at beauty spot Life-threatening cantaloupe recall in four states upgraded to FDA's highest risk level... 'reasonable probability of death' Space scientists spot a'sea slug'... can you see it? Space scientists have released a jaw-dropping image of a space'sea slug'. The stunning picture was taken by NASA's Hubble space telescope which is celebrating it's 36th birthday this week. Situated about 5,000 light-years from Earth, is a star-formation region called the Trifid Nebula which Hubble captured in all its cosmic wonder.

Why did Trump fire Pam Bondi? Trump's US Fed nominee Warsh vows independence, says he's no'sock puppet' Kevin Warsh, United States President Donald Trump's pick to lead the Federal Reserve, has addressed concerns about his independence pending his appointment to the bank amid fears that Trump could sway his decisions on monetary policy. On Tuesday, Warsh -- who served on the central bank's Board of Governors from 2006 to 2011 -- faced waves of criticism during a confirmation hearing of the Senate Banking Committee where Democrats voiced concerns about the Fed's independence should he be appointed to lead the organisation. "I do not believe the operational independence of monetary policy is particularly threatened when elected officials -- presidents, senators, or members of the House -- state their views on interest rates," Warsh said. "Monetary policy independence is essential. Monetary policymakers must act in the nation's interest . . . Warsh, 56, also called for "regime change" at the US central bank, including a new approach for controlling inflation and a communications overhaul that may discourage his colleagues from saying too much about the direction of monetary policy. Warsh blamed the central bank for an inflation surge after it slashed interest rates to nearly zero in the wake of the COVID-19 pandemic, a move that continues to hurt US households. Concerned by the implications of artificial intelligence for jobs - expected to increase productivity - and prices, he said he would move quickly to see if new data tools could provide better insight on inflation, and would also discourage policymakers from saying too much about where interest rates might be heading. "What the Fed needs are reforms to its frameworks and reforms to its communications," the former Fed governor said. "Too many Fed officials opine about where interest rates should be That is quite unhelpful." Warsh has also long been an advocate for shrinking the Fed's $6.7 trillion balance sheet. In the Tuesday hearing, he said any such plans would take time and must be publicly discussed well in advance. Jai Kedia, a research fellow at the Center for Monetary and Financial Alternatives at the libertarian Cato Institute, told Al Jazeera that there were many "encouraging" signs in Warsh's candidacy. "Warsh is presenting himself as a regime change candidate at a time when the Fed needs serious reform," Kedia noted. "Particularly encouraging was his understanding of the negative effects of QE and his focus on reducing the balance sheet.

OpenAI is facing a criminal investigation in the US over whether its ChatGPT technology played a part in the murder of two people during a mass shooting at Florida State University last year. Florida's Attorney General James Uthmeier said on Tuesday his office had been looking into the use of the artificial intelligence (AI) chatbot by a man who allegedly shot several people at the campus in Tallahassee. Our review has revealed that a criminal investigation is necessary, Uthmeier said. ChatGPT offered significant advice to this shooter before he committed such heinous crimes. An OpenAI spokesperson said: ChatGPT is not responsible for this terrible crime.