Angel Reese thanks WNBA for suspending Sandy Brondello, frames'protected species' comment as discrimination Rory McIlroy should be the last golfer to criticize others for'performative' behavior Brexton Busch wins first race since father Kyle Busch's death in emotional return to Victory Lane Caitlin Clark'cheering hard' for Argentina over Spain in World Cup final: 'I want Messi' Ella Langley turns 63-year-old former NFL head coach into'Ella Fella' I am once again begging the folks in USC's athletic department to study some Greek literature Jaxson Dart's swimsuit model girlfriend gets patriotic, golfer Hailey Ostrom takes on Lake Powell & Bigfoot Caitlin Clark's former teammate calls out WNBA for suspending Tempo head coach over'protected species' remark Mark Levin: Trump is the ONLY president to recognize this... Iranian regime has proven they will not abide by'any agreement': Former Israeli ambassador Christopher Nolan brings'The Odyssey' to life with groundbreaking IMAX technology WATCH: Anchor STUNNED by socialist's answer More than 200 military musicians bring America's story to life The World Cup final on Sunday provided a tense, low-scoring finish to a memorable tournament held across the United States. Spain beat Argentina 1-0 in extra time to win the World Cup, but President Donald Trump ended up sharing plenty of the spotlight after joining the trophy ceremony. Rodri of Spain is presented with the FIFA World Cup Winner's Trophy by FIFA President Gianni Infantino and President Donald Trump after Spain's victory over Argentina in the FIFA World Cup 2026 final at New York New Jersey Stadium in East Rutherford, N.J., on July 19, 2026. FIFA President Gianni Infantino and President Donald J. Trump present the FIFA World Cup trophy to Rodri of Spain during the FIFA World Cup 2026 final match between Spain and Argentina at New York New Jersey Stadium in East Rutherford, N.J. (Europa Press Sports/Europa Press) Trump, alongside FIFA president Gianni Infantino, walked to the stage, drawing a mix of cheers and boos as he greeted Spain's players before the trophy presentation. Fox WORLD CUP ANNOUNCER SAYS TRUMP'S APPEARANCE AT FINAL IS'FUN, UNIQUE THING' DESPITE POTENTIAL BACKLASH Trump was briefly part of Spain's team photo as the players hoisted the trophy before Infantino stepped in and suggested he move aside.

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Generating samples from a continuous probability density is a central algorithmic problem across statistics, engineering, and the sciences. For high-dimensional settings, Hamiltonian Monte Carlo (HMC) is the default algorithm across mainstream software packages. However, despite the extensive line of work on HMC and its widespread empirical success, it remains unclear how many iterations of HMC are required as a function of the dimension $d$. On one hand, a variety of results show that Metropolized HMC converges in $O(d^{1/4})$ iterations from a warm start close to stationarity. On the other hand, Metropolized HMC is significantly slower without a warm start, e.g., requiring $Ω(d^{1/2})$ iterations even for simple target distributions such as isotropic Gaussians. Finding a warm start is therefore the computational bottleneck for HMC. We resolve this issue for the well-studied setting of sampling from a probability distribution satisfying strong log-concavity (or isoperimetry) and third-order derivative bounds. We prove that \emph{non-Metropolized} HMC generates a warm start in $\tilde{O}(d^{1/4})$ iterations, after which we can exploit the warm start using Metropolized HMC. Our final complexity of $\tilde{O}(d^{1/4})$ is the fastest algorithm for high-accuracy sampling under these assumptions, improving over the prior best of $\tilde{O}(d^{1/2})$. This closes the long line of work on the dimensional complexity of MHMC for such settings, and also provides a simple warm-start prescription for practical implementations.

This paper considers a non-standard problem of generating samples from a low-temperature Gibbs distribution with \emph{constrained} support, when some of the coordinates of the mode lie on the boundary. These coordinates are referred to as the non-regular part of the model. We show that in a ``pre-asymptotic'' regime in which the limiting Laplace approximation is not yet valid, the low-temperature Gibbs distribution concentrates on a neighborhood of its mode. Within this region, the distribution is a bounded perturbation of a product measure: a strongly log-concave distribution in the regular part and a one-dimensional exponential-type distribution in each coordinate of the non-regular part. Leveraging this structure, we provide a non-asymptotic sampling guarantee by analyzing the spectral gap of Langevin dynamics. Key examples of low-temperature Gibbs distributions include Bayesian posteriors, and we demonstrate our results on three canonical examples: a high-dimensional logistic regression model, a Poisson linear model, and a Gaussian mixture model.

Motivated by applications of large embedding models, we study differentially private (DP) optimization problems under sparsity of individual gradients. We start with new near-optimal bounds for the classic mean estimation problem but with sparse data, improving upon existing algorithms particularly for the high-dimensional regime.

Modifications to a neural network's input and output layers are often required to accommodate the specificities of most practical learning tasks.