$U$-statistics play a central role in statistical inference. In many modern applications, however, acquiring the labels required for $U$-statistics is costly. Motivated by recent advances in active inference, we develop an active inference framework for $U$-statistics that selectively queries informative labels to improve estimation efficiency under a fixed labeling budget, while preserving valid statistical inference. Our approach is built on the augmented inverse probability weighting $U$-statistic, which is designed to incorporate the sampling rule and machine learning predictions. We characterize the optimal sampling rule that minimizes its variance and design practical sampling strategies. We further extend the framework to $U$-statistic-based empirical risk minimization. Experiments on real datasets demonstrate substantial gains in estimation efficiency over baseline methods, while maintaining target coverage.

Neural operators provide a framework for learning solution operators of partial differential equations (PDEs), enabling efficient surrogate modeling for complex systems. While universal approximation results are now well understood, approximation analysis specific to nonlinear reaction-diffusion systems remains limited. In this paper, we study neural operators applied to the solution mapping from initial conditions to time-dependent solutions of a generalized Gierer-Meinhardt reaction-diffusion system, a prototypical model of nonlinear pattern formation. Our main results establish explicit approximation error bounds in terms of network depth, width, and spectral rank by exploiting the Laplacian spectral representation of the Green's function underlying the PDE. We show that the required parameter complexity grows at most polynomially with respect to the target accuracy, demonstrating that Laplacian eigenfunction-based neural operator architectures alleviate the curse of parametric complexity encountered in generic operator learning. Numerical experiments on the Gierer-Meinhardt system support the theoretical findings.

Information-theoretic generalization bounds based on the supersample construction are a central tool for algorithm-dependent generalization analysis in the batch i.i.d.~setting. However, existing supersample conditional mutual information (CMI) bounds do not directly apply to sequential decision-making problems such as online learning, streaming active learning, and bandits, where data are revealed adaptively and the learner evolves along a causal trajectory. To address this limitation, we develop a sequential supersample framework that separates the learner filtration from a proof-side enlargement used for ghost-coordinate comparisons. Under a row-wise exchangeability assumption, the sequential generalization gap is controlled by sequential CMI, a sum of roundwise selector--loss information terms. We also establish a Bernstein-type refinement that yields faster rates under suitable variance conditions. The selector-SCMI proof strategy applies to online learning, streaming active learning with importance weighting, and stochastic multi-armed bandits.

We introduce Pion, a spectrum-preserving optimizer for large language model (LLM) training based on orthogonal equivalence transformation. Unlike additive optimizers such as Adam and Muon, Pion updates each weight matrix through left and right orthogonal transformations, preserving its singular values throughout training. This yields an optimization mechanism that modulates the geometry of weight matrices while keeping their spectral norm fixed. We derive the Pion update rule, systematically examine its design choices, and analyze its convergence behavior along with several key properties. Empirical results show that Pion offers a stable and competitive alternative to standard optimizers for both LLM pretraining and finetuning.

Chinese firm unveils'transformer' style manned robot NewsFeed Chinese firm unveils'transformer' style manned robot Unitree Robotics has released footage of its CEO piloting the GD01, a 2.7-metre transformable mecha that smashes through walls with its mechanical arms. The machine is priced from $650,000. Starmer at risk because he pushed Labour to be'new Conservative Party' Trump skirts question on US'red lines' for Iran ceasefire How one Palestinian teen's life changed forever after Israeli gunfire

If You Have $650,000 and Don't Buy This Giant Mecha Robot You're a Fool China's Unitree, famous for making low-cost dancing robots, will now sell you a giant, wall-smashing mecha. Unitree is a Chinese company known for making adorable, relatively affordable robots that dance and shuffle and such. Last night, it revealed its latest creation, which is something of a departure: a giant, walking, crawling, transforming, wall-smashing "mecha" called the GD01. An introductory video for the GD01--set to a thundering rock guitar soundtrack--shows the company's founder and CEO, Xingxing Wang, holding hands with the robot before climbing into its prodigious, open-air belly. A disclaimer added to Unitree's social media post reads: "Please everyone be sure to use the robot in a Friendly and Safe manner."

When China's leader Xi Jinping hosts his American counterpart in Beijing this week, Donald Trump will be reminded of his last visit in 2017 - he was wooed hard, complete with dinner inside the Forbidden City, an honour no US president before him had received. This week's reception promises to be just as grand, including a stop inside Zhongnanhai, the rarefied compound where China's top leadership lives and works. The agenda too will be just as thorny, with Iran being a new source of tension, alongside trade, technology and Taiwan. But a lot has changed as Trump returns to a stronger and far more assertive China. Now well into an unprecedented third term, an ambitious Xi has been pushing forward with plans for new productive forces with heavy investments in renewable energy, robotics and artificial intelligence.

Married doctor's affair with glamorous younger woman explodes into Fatal Attraction-style court war... X-rated photo claims, leaked recordings and a sinister threat: 'I'll never stop' NBA rocked as Grizzlies star Brandon Clarke dies suddenly at 29... a month after being arrested on drug charges The unassuming apps all cheaters use to hide their affairs: Where to look on your partner's phone to see exactly what they are up to... and the subtle red flags to never ignore I've treated so many cocaine users. This is the one sign that makes it so obvious you have a problem, how it can kill you in a night... and the embarrassing sexual side effect you may not have heard of: DR PHILIPPA KAYE Explosive Supreme Court LEAK reveals stinging whispers about'belligerent' justice read the wild rants troubling both sides of the aisle Surge in cancer patients taking 20 cent'wonder drug' after Mel Gibson claims that friends beat incurable disease thanks to drug The'marry me' sex move that'll make even the most commitment-phobic of men beg to see you again... and it worked for THREE of my friends Trump's chilling'treason' note revealed as he hunts down Iran war leakers... and Israel bombshell sparks fury Hollywood's $350k matchmaker exposes the secret love lives of the rich and famous: Diva demands, fake names, NDAs... and how to know if your relationship is doomed Secret trove of injury photos that blow apart married tech mogul's family-man image revealed in explosive lawsuit: Bruises, beatings and forced sex acts he allegedly inflicted on girlfriend Furious argument explodes on CNN after panelist flagged Kevin O'Leary's old age during foul-mouthed fight about politics He knew Elizabeth Taylor's secrets. Johnny Depp came to him for answers. But Hollywood's greatest confidante buried a betrayal that was too dangerous to expose The unassuming apps all cheaters use to hide their affairs: Where to look on your partner's phone to see exactly what they are up to... and the subtle red flags to never ignore The same apps millions use to share music, split bills and track workouts are now being quietly repurposed by cheaters to hide secret relationships in plain sight. Experts warn modern affairs are increasingly unfolding inside ordinary platforms that appear so harmless that most partners would never think twice about them.

The UK has said it will contribute drones, fighter jets and a warship to a joint mission aimed at safeguarding shipping in the Strait of Hormuz. Defence minister John Healey announced the package at a virtual summit of defence ministers on Tuesday. It includes autonomous systems to detect and clear naval mines, drone boats and Typhoon jets for air patrols. More than 40 other nations are involved in the mission, which Healey said would begin when conditions allow. For months Iran has been controlling the Strait of Hormuz - one of the world's busiest oil shipping channels - in retaliation for US and Israeli attacks.

Iran Is Using Tiny'Mosquito' Boats to Shut Down the Strait of Hormuz Iran's traditional naval fleet has been almost completely destroyed by US-Israeli raids. But Iran's Revolutionary Guard has deployed a fleet of small vessels that is crippling every passageway. In the Strait of Hormuz, Iran has developed an asymmetrical naval strategy that is crippling the passage of container ships. This "hemostat" uses guerrilla tactics, after Iran's "traditional" fleet was almost entirely destroyed by US and Israeli attacks. No longer able to rely on specialized military ships, Tehran is using an unconventional force made up of dozens of small military vessels armed with missiles, machine guns, and drones.