Under the assumption that data lie on a compact (unknown) manifold without boundary, we derive finite sample bounds for kernel smoothing and its (first and second) derivatives, and we establish asymptotic normality through Berry-Esseen type bounds. Special cases include kernel density estimation, kernel regression and the heat kernel signature. Connections to the graph Laplacian are also discussed.

Polynomial Chaos Expansions (PCEs) are widely recognized for their efficient computational performance in surrogate modeling. Yet, a robust framework to quantify local model errors is still lacking. While the local uncertainty of PCE prediction can be captured using bootstrap resampling, other methods offering more rigorous statistical guarantees are needed, especially in the context of small training datasets. Recently, conformal predictions have demonstrated strong potential in machine learning, providing statistically robust and model-agnostic prediction intervals. Due to its generality and versatility, conformal prediction is especially valuable, as it can be adapted to suit a variety of problems, making it a compelling choice for PCE-based surrogate models. In this contribution, we explore its application to PCE-based surrogate models. More precisely, we present the integration of two conformal prediction methods, namely the full conformal and the Jackknife+ approaches, into both full and sparse PCEs. For full PCEs, we introduce computational shortcuts inspired by the inherent structure of regression methods to optimize the implementation of both conformal methods. For sparse PCEs, we incorporate the two approaches with appropriate modifications to the inference strategy, thereby circumventing the non-symmetrical nature of the regression algorithm and ensuring valid prediction intervals. Our developments yield better-calibrated prediction intervals for both full and sparse PCEs, achieving superior coverage over existing approaches, such as the bootstrap, while maintaining a moderate computational cost.

Weather conditions can drastically alter the state of crops and rangelands, and in turn, impact the incomes and food security of individuals worldwide. Satellite-based remote sensing offers an effective way to monitor vegetation and climate variables on regional and global scales. The annual peak Normalized Difference Vegetation Index (NDVI), derived from satellite observations, is closely associated with crop development, rangeland biomass, and vegetation growth. Although various machine learning methods have been developed to forecast NDVI over short time ranges, such as one-month-ahead predictions, long-term forecasting approaches, such as one-year-ahead predictions of vegetation conditions, are not yet available. To fill this gap, we develop a two-phase machine learning model to forecast the one-year-ahead peak NDVI over high-resolution grids, using the Four Corners region of the Southwestern United States as a testbed. In phase one, we identify informative climate attributes, including precipitation and maximum vapor pressure deficit, and develop the generalized parallel Gaussian process that captures the relationship between climate attributes and NDVI. In phase two, we forecast these climate attributes using historical data at least one year before the NDVI prediction month, which then serve as inputs to forecast the peak NDVI at each spatial grid. We developed open-source tools that outperform alternative methods for both gross NDVI and grid-based NDVI one-year forecasts, providing information that can help farmers and ranchers make actionable plans a year in advance.

We study a general framework of distributional computational graphs: computational graphs whose inputs are probability distributions rather than point values. We analyze the discretization error that arises when these graphs are evaluated using finite approximations of continuous probability distributions. Such an approximation might be the result of representing a continuous real-valued distribution using a discrete representation or from constructing an empirical distribution from samples (or might be the output of another distributional computational graph). We establish non-asymptotic error bounds in terms of the Wasserstein-1 distance, without imposing structural assumptions on the computational graph.

Language diffusion models aim to improve sampling speed and coherence over autoregressive LLMs. We introduce Neural Flow Diffusion Models for language generation, an extension of NFDM that enables the straightforward application of continuous diffusion models to discrete state spaces. NFDM learns a multivariate forward process from the data, ensuring that the forward process and generative trajectory are a good fit for language modeling. Our model substantially reduces the likelihood gap with autoregressive models of the same size, while achieving sample quality comparable to that of previous latent diffusion models.

This paper investigates two fundamental descriptors of data, i.e., density distribution versus mass distribution, in the context of clustering. Density distribution has been the de facto descriptor of data distribution since the introduction of statistics. We show that density distribution has its fundamental limitation -- high-density bias, irrespective of the algorithms used to perform clustering. Existing density-based clustering algorithms have employed different algorithmic means to counter the effect of the high-density bias with some success, but the fundamental limitation of using density distribution remains an obstacle to discovering clusters of arbitrary shapes, sizes and densities. Using the mass distribution as a better foundation, we propose a new algorithm which maximizes the total mass of all clusters, called mass-maximization clustering (MMC). The algorithm can be easily changed to maximize the total density of all clusters in order to examine the fundamental limitation of using density distribution versus mass distribution. The key advantage of the MMC over the density-maximization clustering is that the maximization is conducted without a bias towards dense clusters.

Prince Harry and Meghan Markle's Sundance screening sparks online row: 'Sussex Squad' brand claims event failed to sell out as'lies' despite photos showing'rows of empty seats' Mick Jagger's family launch desperate hunt for missing relative: His granddaughter's partner vanishes in Cornwall after wandering streets Forensic video analysis of Alex Pretti's final 30 seconds exposes'John Wayne gun' question that can't be ignored Sinister truth about Celine Dion's song All By Myself: Singer's producer reveals bombshell secrets of her 26-year age gap marriage... that he swore not to tell until her husband René died The nastiest clique in Hollywood have had their dirty secret outed... there's no coming back from this: MAUREEN CALLAHAN Ariana Grande and Cynthia Erivo'creeped a lot of people out' says anonymous Oscar voter amid Wicked snubs John Fetterman's own WIFE turns on him over ICE as Senator comes under fire for his silence on shooting of Alex Pretti Lauren Sanchez turns heads in a red skirt suit as she holds hands with billionaire husband Jeff Bezos at Schiaparelli's Paris Haute Couture Fashion Week show Olivia Wilde blasts'inauthentic and unrealistic' sex in modern film and claims it has'been that way for a long time' - despite featuring racy scenes in Don't Worry Darling Sandra Bullock's Blind Side costar Quinton Aaron is'fighting for his life' in hospital after falling at home Seedy underbelly of America's exclusive golf clubs... as cart girls expose ultra-rich world of sex scandals and drunken debauchery Real estate mogul is sensationally found GUILTY of murdering football coach's son outside mall Kelly Clarkson on verge of QUITTING: Staff are all starting to say same thing backstage... as friends let slip the only way she could be convinced to stay Panicking realtors are drowning in unsold homes in America's'most extreme' market. They blame'the Joe Rogan effect' Father of alien archaeology says the pyramids were not built by human hands... and claims he has proof READ MORE: Egypt's Great Pyramid construction rewritten as new evidence exposes how it was actually built The belief that the pyramids were not built by human hands has fascinated conspiracy theorists for decades. No one promoted that idea more persistently than Swiss author Erich von Däniken, often described as the father of ancient alien archaeology. Von Däniken, who died this month aged 90, argued that extraterrestrial visitors played a direct role in helping ancient Egyptians construct monuments that would otherwise have been impossible. In his 1968 bestseller'Chariots of the Gods,' he claimed alien'astronauts' visited early civilizations, including the ancient Egyptians and Mayans, and shared advanced technology.

Environment Animals Wildlife Fish'Walking sharks' lay eggs without breaking a sweat Breakthroughs, discoveries, and DIY tips sent six days a week. Being pregnant and giving birth is hard work for any species--but epaulette sharks () might disagree. These fish and a number of other species are known as " walking sharks " for their ability to traverse both the seafloor and land with their fins. Epaulette sharks' energy use didn't change during their reproduction cycle, as described in a study recently published in the journal . "Reproduction is the ultimate investment you are literally building new life from scratch," Jodie Rummer, a marine biologist at James Cook University and co-author of the recent study, said in a university statement .

Shocking history of handgun Alex Pretti was carrying when he was shot by Border Patrol is revealed... as judge bans Trump administration from'destroying evidence' from scene Homeowners in PANIC as stark map shows just five sellers' markets left - spelling price drops almost everywhere else My secret fantasy appalled my husband... and it has revealed an irreparable rift in our marriage: DEAR JANE Seedy underbelly of America's exclusive golf clubs... as cart girls expose ultra-rich world of sex scandals and drunken debauchery Doomsday Clock will be updated next WEEK to determine our fate - here's how scientists think the hands will move Lisa Rinna's nepo baby daughter Amelia Gray Hamlin, 24, breaks down all the plastic surgery she's had Gay reporter's wild on-air comment about Patriots quarterback Drake Maye goes viral Meteorologist reveals America's most dangerous city in winter storm's corridor of chaos: 'Staying in your home won't be viable' '90s bombshell channels iconic role she took over from Pamela Anderson and looks exactly the same at 59 Crystal clear new video raises horrifying questions about killing of Minneapolis nurse by DHS...as doctor on scene claims agents were counting bullet holes instead of helping him Doomsday Clock will be updated next WEEK to determine our fate - here's how scientists think the hands will move Humanity is about to learn if we have moved closer to self-destruction as the Doomsday Clock is updated. The new time for the symbolic timepiece, which ticks closer to midnight as we approach annihilation, will be revealed on Tuesday, January 27. Since last year, the clock has sat at 89 seconds to midnight - the latest time in its 78-year history. However, experts have told the Daily Mail they now expect the Doomsday Clock to move even closer to midnight . While the Doomsday Clock was initially created to track the risk of nuclear war between Russia and America, the world now faces a far more diverse array of threats.

As many as 75,000 mating pairs are waiting for eggs. Breakthroughs, discoveries, and DIY tips sent six days a week. While winter is raging in an unusually large swath of the United States, the weather is balmy for the birds nesting on the Pacific Ocean's Midway Atoll. As many as 75,000 pairs of Laysan albatrosses (or mōlī in Hawaiian) are nesting in the wildlife refuge on the northwestern edge of the Hawaiian Archipelago. Now you can watch these brilliant snow-white birds while avoiding the actual snow with a 24/7 live cam.