In this paper, we investigate the supremum-norm generalization error and the uniform inference for a specific class of kernel regression methods, namely the kernel gradient flows. Under the widely adopted capacity-source condition framework in the kernel regression literature, we first establish convergence rates for the supremum norm generalization error of both continuous and discrete kernel gradient flows under the source condition $s>α_0$, where $α_0\in(0,1)$ denotes the embedding index of the kernel function. Moreover, we show that these rates match the minimax optimal rates. Building on this result, we then construct simultaneous confidence bands for both continuous and discrete kernel gradient flows. Notably, the widths of the proposed confidence bands are also optimal, in the sense that their shrinkage rates are greater than, while can be arbitrarily close to, the minimax optimal rates.

Trimming suspicious calibration points is a common response to contamination in conformal prediction. Its effect on clean-target coverage, however, is governed by the retained law induced by trimming, not by the contamination level alone. We analyse fixed-threshold trimming as conditioning rather than purification. It replaces the contaminated calibration law with a retained law, reducing clean-target coverage to a one-dimensional score-CDF transfer problem with an exact finite-sample identity. A componentwise bound on the transfer gap gives a population-level diagnostic. This separates a clean-side covariance cost from a retained-contamination cost, governed by the dirty-to-clean retention ratio. Trimming helps when the anomaly score separates retention probabilities while remaining score-neutral on the clean population. Otherwise, it cannot substantially reduce contamination through the retained mixture coefficient. We also give finite-sample certificate templates that provide numerical guarantees under independent audit.

Despite the prevalence of the attention sink phenomenon in Large Language Models (LLMs), where initial tokens disproportionately monopolize attention scores, its structural origins remain elusive. This work provides a \textit{mechanistic explanation} for this phenomenon. First, we trace its root to the value aggregation process inherent in self-attention, which induces a systematic variance discrepancy. We further demonstrate that this discrepancy is drastically amplified by the activation of super neurons within Feed-Forward Network (FFN) layers. Specifically, the channel-sparse down-projections trigger a dimension disparity of the first-token representation, necessitating the formation of attention sinks as a structural anchor. Then, we validate this causal chain through two controlled interventions: (i) isolating the aggregation effect via attention mask modifications and (ii) amplifying the variance of targeted token representations. Both interventions can replicate attention sinks at arbitrary positions. Our mechanistic understanding offers a foundation for the systematic control of sink formation. Finally, as a proof of concept, we propose \textit{head-wise RMSNorm}, an architectural modification that stabilizes value aggregation outputs during pre-training. Our experiments demonstrate that restoring statistical parity across positions significantly accelerates convergence.

Here's why the coaches association's 24-team College Football Playoff could ruin the sport Boston Celtics star Jaylen Brown tells ESPN's Stephen A Smith to'be quiet and retire' President Trump on $1,000 World Cup ticket prices: 'I wouldn't pay it either, to be honest' Pirates vs. Diamondbacks betting preview targets the under as both offenses go cold in series Former LSU coach Brian Kelly uses AI to prepare for job interviews, proving he's just like the rest of us Newsom office source responds to planned protest against trans athlete at state playoff girls' track meet'This can touch anyone': Gorman family speaks following loss of Sheridan'Project Freedom' could soon resume: Report Iranian people are not citizens, but'subjects' of the regime: Middle East expert Vice Admiral Robert Harward weighs in on restarting'Project Freedom' in Strait of Hormuz Largest teachers' union accused of antisemitism in federal civil rights complaint McEnany's URGENT plea: 'Be Spencer Pratt!' WHO doesn't expect large Hantavirus outbreak US blockade keeps stranglehold on Iran's economy What And Who Is Next In The Russini Saga? | OutKick Hot Mic TMZ Sports obtained documents this week showing that New England Patriots coach Mike Vrabel and former NFL reporter Dianna Russini signed a waiver for a private boating trip during her pregnancy in 2021. The report adds that Vrabel and Russini were the only two people on board for the two-to-three-hour excursion. On Thursday, the outlet obtained video and photos of the two on a dock in Putnam County, TN. See TMZ Sports' video below: ARE WE SURE MIKE VRABEL WILL SURVIVE RUSSINI SCANDAL AND COACH PATRIOTS THIS SEASON? According to the report, Russini gave birth to her first child with her husband, Kevin, later that summer. She was about six to seven months pregnant during the boating trip with Vrabel.

Boston Celtics star Jaylen Brown tells ESPN's Stephen A Smith to'be quiet and retire' President Trump on $1,000 World Cup ticket prices: 'I wouldn't pay it either, to be honest' Pirates vs. Diamondbacks betting preview targets the under as both offenses go cold in series Former LSU coach Brian Kelly uses AI to prepare for job interviews, proving he's just like the rest of us Newsom office source responds to planned protest against trans athlete at state playoff girls' track meet'This can touch anyone': Gorman family speaks following loss of Sheridan'Project Freedom' could soon resume: Report Iranian people are not citizens, but'subjects' of the regime: Middle East expert Vice Admiral Robert Harward weighs in on restarting'Project Freedom' in Strait of Hormuz Largest teachers' union accused of antisemitism in federal civil rights complaint McEnany's URGENT plea: 'Be Spencer Pratt!' WHO doesn't expect large Hantavirus outbreak US blockade keeps stranglehold on Iran's economy OutKick Here's why the coaches association's 24-team College Football Playoff could ruin the sport College Football Playoff Eying 24 Team Expansion: Is Bigger Better? The college football playoff could expand to 24 teams -- and a lot of people aren't happy about it. After just moving to 12, this feels like overkill. Is the sport trying to fix something that wasn't broken? When college football first moved to a playoff format, there were just four teams included.

Here's why the coaches association's 24-team College Football Playoff could ruin the sport President Trump on $1,000 World Cup ticket prices: 'I wouldn't pay it either, to be honest' Pirates vs. Diamondbacks betting preview targets the under as both offenses go cold in series Former LSU coach Brian Kelly uses AI to prepare for job interviews, proving he's just like the rest of us Newsom office source responds to planned protest against trans athlete at state playoff girls' track meet Iranians are fearful of facing the regime's frustration and anger after the war, activist says'This can touch anyone': Gorman family speaks following loss of Sheridan'Project Freedom' could soon resume: Report Iranian people are not citizens, but'subjects' of the regime: Middle East expert Vice Admiral Robert Harward weighs in on restarting'Project Freedom' in Strait of Hormuz Largest teachers' union accused of antisemitism in federal civil rights complaint McEnany's URGENT plea: 'Be Spencer Pratt!' WHO doesn't expect large Hantavirus outbreak OutKick Boston Celtics star Jaylen Brown tells ESPN's Stephen A Smith to'be quiet and retire' The viral exchange on X adds Brown to a list of NBA stars, including LeBron James and Kevin Durant, who've feuded with Smith ESPN commentator Stephen A. Smith is no stranger to having beef with NBA stars. It's time to add Celtics guard Jaylen Brown to the list. The latest dust-up started, naturally, on First Take, where Smith took aim at Brown for his comments following Boston's playoff collapse . Brown recently said this was his favorite year, despite the Celtics blowing a 3-1 series lead to the Philadelphia 76ers and getting eliminated in the first round of the NBA playoffs. That didn't sit well with Smith, who made it very clear Thursday that he thought Brown should have kept that to himself.

Days before the cruise ship affected by a hantavirus outbreak is set to dock in southern Tenerife in the Canary Islands, locals are expressing a mix of concern and calm. According to Oceanwide Expeditions, there are currently no symptomatic individuals on board the ship. Iran's president says he has spoken to the country's Supreme Leader

President Trump on $1,000 World Cup ticket prices: 'I wouldn't pay it either, to be honest' Pirates vs. Diamondbacks betting preview targets the under as both offenses go cold in series Former LSU coach Brian Kelly uses AI to prepare for job interviews, proving he's just like the rest of us Newsom office source responds to planned protest against trans athlete at state playoff girls' track meet Framber Valdez gets what he deserves for punk move, suspended six games after drilling Boston's Trevor Story'Project Freedom' could soon resume: Report Iranian people are not citizens, but'subjects' of the regime: Middle East expert Vice Admiral Robert Harward weighs in on restarting'Project Freedom' in Strait of Hormuz Largest teachers' union accused of antisemitism in federal civil rights complaint McEnany's URGENT plea: 'Be Spencer Pratt!' WHO doesn't expect large Hantavirus outbreak US blockade keeps stranglehold on Iran's economy The UAB athletic department is investigating allegations of mental and physical abuse of players by head softball coach Taylor Smartt. Entering her second season with the Blazers in Birmingham, Alabama, Smartt had hoped to turn her first stint as a head coach into a success story. But there had been discussions around her coaching philosophy just months after she set foot on campus. Now, nearly two years later, the school finds itself in a terrible spot, as accusations of serious misconduct arise and an investigation by the athletic department heats up. On April 25, Taylor stepped away from the UAB softball team as the team was headed to Florida for a three-game series with USF in the American Conference.

'This is an apartheid regime' A Palestinian boy mourns his father, one of three people killed in an Israeli strike on a security post in Gaza. The attack is part of ongoing Israeli violence, despite a fragile ceasefire, which has killed at least 846 Palestinians, according to Gaza's Health Ministry. Iran's president says he has spoken to the country's Supreme Leader

Here's What You Need to Know About the Hantavirus While the outbreak aboard a cruise ship in the Atlantic is concerning, the virus isn't easily transmitted through casual contact. Cruises are so closely associated with illness that the highly contagious norovirus is commonly called the "cruise ship virus." But a ship headed for Spain's Canary Islands has attracted global attention due to a rare outbreak of hantavirus that's left three dead. While alarming, health officials and infectious disease experts say the risk to the general public right now is low because hantavirus is less contagious than other respiratory diseases like the coronavirus responsible for the Covid-19 pandemic . "This is not Covid, this is not influenza. It spreads very, very differently," Maria Van Kerkhove, director of epidemic and pandemic preparedness and prevention at the World Health Organization, said at a press conference on Thursday.