You're viewing the US edition You can switch to the UK, AU or IE homepage at any time using this menu. Staggering modern breakthroughs that prove the Bible is true: From Noah's Ark's resting place to bloody clues on the Shroud of Turin, experts reveal evidence that convinced them Paranoid Kimberly Guilfoyle'holed up' inside lavish Athens estate during humiliating demise: Now brazen phone calls and new gambit to pay for luxury G-Wagon are revealed Furious Trump delivers brutal final dig at Kimberly Guilfoyle after dramatic resignation as Greek ambassador: 'I guess she dated my son' Inside RJ Mitte's devastating final days: Tormented Breaking Bad star was'nearly broke'... and thought his fiancée deserved'someone better' JD Vance hints Texas Democrat James Talarico is gay in sly jibe: 'I'm the Vice President, not James Talarico' I had my legs, lips and favorite tattoo amputated after hysterectomy went terribly wrong. Presley Gerber's sober coach revealed for first time: Former addict'going through hell' since dropping off longtime client to rehab where he died... as new details of companion's past are divulged How a drug widely given for restless legs left Dorothy with a compulsion to shop online - and has driven others to unfettered gambling, eating... and sex Amy Robach and T.J. Holmes are married! Kelly Clarkson's furious outburst at staff inside Brooklyn store revealed: Sickly owner makes new claims about star as feud spirals Breaking Bad star RJ Mitte's private agony revealed: Read dozens of devastating messages to friend as Bryan Cranston is left'heartbroken' by actor's death at 34 I'm in my 50s and at least four of my friends are cheating on their husbands. Trump demands answers on Ilhan Omar's $30M and accuses her of being here'illegally' You've shown us how you REALLY feel about women... and it's no joke: JANA HOCKING unleashes on the SNL comedian Grieving Kaia Gerber is seen with parents Cindy Crawford and Rande for first time since Presley's shock death at 27 Trump erupts as'panic' grips White House over stories of his own midterms masterminds cashing in... and Susie Wiles calls crisis meeting The 72 hours of chaos before Trump hired press secretary: Karoline Leavitt's late warning, how the president bypassed Susie Wiles... and the names on the chopping block as Katie Zacharia arrives Jimmy Kimmel calls Trump a'petty little b***h' in foul monologue about president's Nobel Peace Prize tantrum Apple HomeView hands-on: SHIVALI BEST is one of the first people to test the £299 home device - so, should you buy one for your family?

You're viewing the US edition You can switch to the UK, AU or IE homepage at any time using this menu. Leaked security footage casts new light on Kelly Clarkson's furious feud with Brooklyn store Kimberly Guilfoyle's burn book revealed. Her MAGA secrets kept in a journal so potent it has nervous Trump insiders fearing revenge: 'She knows where the bodies are buried' Inside Selma Blair's extraordinary battle with MS: Years of missed symptoms, other highly unusual conditions and the mystery medication blamed for her midair meltdown I was married to a serial killer. These are the signs I missed... and the chilling realization I had when a victim who escaped his torture chamber contacted me decades later I too was gang raped by a group of college students. This is the uncomfortable truth about Cornell's Jane Doe that no woman wants to discuss Paranoid Kimberly Guilfoyle'holed up' inside lavish Athens estate during humiliating demise: Now brazen phone calls and new gambit to pay for luxury G-Wagon are revealed A father's Six Flags death is not the only roller coaster tragedy to strike... it raises chilling questions over theme park safety Trump again promises $5k checks for every American but says you can't go on vacation... as he scolds protestor with brutal zinger Staggering modern breakthroughs that prove the Bible is true: From Noah's Ark's resting place to bloody clues on the Shroud of Turin, experts reveal evidence that convinced them I had my legs, lips and favorite tattoo amputated after hysterectomy went terribly wrong.

We propose a control-theoretic framework for evolutionary clustering based on Mean Field Games (MFG). Moving beyond static or heuristic approaches, we formulate the problem as a population dynamics game governed by a coupled Hamilton-Jacobi-Bellman and Fokker-Planck system. Driven by a variational cost functional rather than predefined statistical shapes, this continuous-time formulation provides a flexible basis for non-parametric cluster evolution. To validate the framework, we analyze the setting of time-dependent Gaussian mixtures, showing that the MFG dynamics recover the trajectories of the classical Expectation-Maximization (EM) algorithm while ensuring mass conservation. Furthermore, we introduce time-averaged log-likelihood functionals to regularize temporal fluctuations. Numerical experiments illustrate the stability of our approach and suggest a path toward more general non-parametric clustering applications where traditional EM methods may face limitations.

Predictive inference is a fundamental task in statistics, traditionally addressed using parametric assumptions about the data distribution and detailed analyses of how models learn from data. In recent years, conformal prediction has emerged as a rapidly growing alternative framework that is particularly well suited to modern applications involving high-dimensional data and complex machine learning models. Its appeal stems from being both distribution-free -- relying mainly on symmetry assumptions such as exchangeability -- and model-agnostic, treating the learning algorithm as a black box. Even under such limited assumptions, conformal prediction provides exact finite-sample guarantees, though these are typically of a marginal nature that requires careful interpretation. This paper explains the core ideas of conformal prediction and reviews selected methods. Rather than offering an exhaustive survey, it aims to provide a clear conceptual entry point and a pedagogical overview of the field.

Classical Gaussian processes and Kriging models are commonly based on stationary kernels, whereby correlations between observations depend exclusively on the relative distance between scattered data. While this assumption ensures analytical tractability, it limits the ability of Gaussian processes to represent heterogeneous correlation structures. In this work, we investigate variably scaled kernels as an effective tool for constructing non-stationary Gaussian processes by explicitly modifying the correlation structure of the data. Through a scaling function, variably scaled kernels alter the correlations between data and enable the modeling of targets exhibiting abrupt changes or discontinuities. We analyse the resulting predictive uncertainty via the variably scaled kernel power function and clarify the relationship between variably scaled kernels-based constructions and classical non-stationary kernels. Numerical experiments demonstrate that variably scaled kernels-based Gaussian processes yield improved reconstruction accuracy and provide uncertainty estimates that reflect the underlying structure of the data

Modern surgery has gone from long incisions to tiny cuts guided by robots and AI. In the process, however, surgeons have lost something vital: the chance to feel inside the body directly. Without palpation, it becomes harder to detect tissue abnormalities during an operation. A group of surgeons and engineers across Europe is now trying to bring back this vital aspect of surgery. Working within an EU-funded research collaboration called PALPABLE, they are developing a soft robotic "fingertip" that can sense how firm or soft tissue is during minimally invasive and robotic surgery.

Heavy-tailed distributions are ubiquitous in real-world data, where rare but extreme events dominate risk and variability. However, standard Variational Autoencoders (VAEs) employ simple decoder distributions (e.g., Gaussian) that fail to capture heavy-tailed behavior, while existing heavy-tail-aware extensions remain restricted to predefined parametric families whose tail behavior is fixed a priori. We propose the Phase-Type Variational Autoencoder (PH-VAE), whose decoder distribution is a latent-conditioned Phase-Type (PH) distribution defined as the absorption time of a continuous-time Markov chain (CTMC). This formulation composes multiple exponential time scales, yielding a flexible and analytically tractable decoder that adapts its tail behavior directly from the observed data. Experiments on synthetic and real-world benchmarks demonstrate that PH-VAE accurately recovers diverse heavy-tailed distributions, significantly outperforming Gaussian, Student-t, and extreme-value-based VAE decoders in modeling tail behavior and extreme quantiles. In multivariate settings, PH-VAE captures realistic cross-dimensional tail dependence through its shared latent representation. To our knowledge, this is the first work to integrate Phase-Type distributions into deep generative modeling, bridging applied probability and representation learning.

Sparse recovery is among the most well-studied problems in learning theory and high-dimensional statistics. In this work, we investigate the statistical and computational landscapes of sparse recovery with $\ell_\infty$ error guarantees. This variant of the problem is motivated by \emph{variable selection} tasks, where the goal is to estimate the support of a $k$-sparse signal in $\mathbb{R}^d$. Our main contribution is a provable separation between the \emph{oblivious} (``for each'') and \emph{adaptive} (``for all'') models of $\ell_\infty$ sparse recovery. We show that under an oblivious model, the optimal $\ell_\infty$ error is attainable in near-linear time with $\approx k\log d$ samples, whereas in an adaptive model, $\gtrsim k^2$ samples are necessary for any algorithm to achieve this bound. This establishes a surprising contrast with the standard $\ell_2$ setting, where $\approx k \log d$ samples suffice even for adaptive sparse recovery. We conclude with a preliminary examination of a \emph{partially-adaptive} model, where we show nontrivial variable selection guarantees are possible with $\approx k\log d$ measurements.

Two key features make correlation clustering quite suitable in real-world applications.