Diffusion models generate samples by iteratively querying learned score estimates. A rapidly growing literature focuses on accelerating sampling by minimizing the number of score evaluations, yet the information-theoretic limits of such acceleration remain unclear. In this work, we establish the first score query lower bounds for diffusion sampling. We prove that for $d$-dimensional distributions, given access to score estimates with polynomial accuracy $\varepsilon=d^{-O(1)}$ (in any $L^p$ sense), any sampling algorithm requires $\widetildeΩ(\sqrt{d})$ adaptive score queries. In particular, our proof shows that any sampler must search over $\widetildeΩ(\sqrt{d})$ distinct noise levels, providing a formal explanation for why multiscale noise schedules are necessary in practice.

While the manifold hypothesis is widely adopted in modern machine learning, complex data is often better modeled as stratified spaces -- unions of manifolds (strata) of varying dimensions. Stratified learning is challenging due to varying dimensionality, intersection singularities, and lack of efficient models in learning the underlying distributions. We provide a deep generative approach to stratified learning by developing two generative frameworks for learning distributions on stratified spaces. The first is a sieve maximum likelihood approach realized via a dimension-aware mixture of variational autoencoders. The second is a diffusion-based framework that explores the score field structure of a mixture. We establish the convergence rates for learning both the ambient and intrinsic distributions, which are shown to be dependent on the intrinsic dimensions and smoothness of the underlying strata. Utilizing the geometry of the score field, we also establish consistency for estimating the intrinsic dimension of each stratum and propose an algorithm that consistently estimates both the number of strata and their dimensions. Theoretical results for both frameworks provide fundamental insights into the interplay of the underlying geometry, the ambient noise level, and deep generative models. Extensive simulations and real dataset applications, such as molecular dynamics, demonstrate the effectiveness of our methods.

Since their introduction by Breiman (2001), permutation-based feature importance measures have been widely adopted. However, randomly permuting the entries of a dataset may create new points far from the original data or even "impossible data." In a permuted dataset, we may find children who are retired or individuals who graduated from high school before they were born (Mase et al. 2022, p. 1). Forcing ML models to make predictions at these points causes them to extrapolate, making explanations unreliable (Hooker et al. 2021). Every non-trivial permutation-based variable importance measure, including SHAP (Lundberg and Lee 2017), Knockoffs (Barber and Candés 2015), conditional model reliance (Fisher et al. 2019), and accumulated local effect (ALE) plots (Apley and Zhu 2020) suffer from this. We propose and compare three new strategies to address extrapolation issues. The first combines conditional model reliance from Fisher et al. (2019) with a Gaussian transformation. By mapping data quantiles to a Gaussian distribution and back, we adjust only the quantiles of point values, significantly reducing extrapolation. Under a Gaussian copula assumption for the feature distribution, we prove that the new data points follow the same probability distribution as the original data.

Generalized linear mixed-effects models (GLMMs) are widely used to analyze grouped and hierarchical data. In a GLMM, each response is assumed to follow an exponential-family distribution where the natural parameter is given by a linear function of observed covariates and a latent group-specific random effect. Since exact marginalization over the random effects is typically intractable, model parameters are estimated by maximizing an approximate marginal likelihood. In this paper, we replace the linear function with neural networks. The result is a more flexible model, the neural generalized mixed-effects model (NGMM), which captures complex relationships between covariates and responses. To fit NGMM to data, we introduce an efficient optimization procedure that maximizes the approximate marginal likelihood and is differentiable with respect to network parameters. We show that the approximation error of our objective decays at a Gaussian-tail rate in a user-chosen parameter. On synthetic data, NGMM improves over GLMMs when covariate-response relationships are nonlinear, and on real-world datasets it outperforms prior methods. Finally, we analyze a large dataset of student proficiency to demonstrate how NGMM can be extended to more complex latent-variable models.

In many problems of data-driven modeling for dynamical systems, the governing equations are not known a priori and must be selected phenomenologically from a large set of candidate interactions and basis functions. In such situations, point estimates alone can be misleading, because multiple model components may explain the observed data comparably well, especially when the data are limited or the dynamics exhibit poor identifiability. Quantifying the uncertainty associated with model selection is therefore essential for constructing reliable dynamical models from data. In this work, we develop a Bayesian sparse identification framework for dynamical systems with coupled components, aimed at inferring both interaction structure and functional form together with principled uncertainty quantification. The proposed method combines sparse modeling with Bayesian model averaging, yielding posterior inclusion probabilities that quantify the credibility of each candidate interaction and basis component. Through numerical experiments on oscillator networks, we show that the framework accurately recovers sparse interaction structures with quantified uncertainty, including higher-order harmonic components, phase-lag effects, and multi-body interactions. We also demonstrate that, even in a phenomenological setting where the true governing equations are not contained in the assumed model class, the method can identify effective functional components with quantified uncertainty. These results highlight the importance of Bayesian uncertainty quantification in data-driven discovery of dynamical models.

Valuable coins including a gold rush era "Humbert Slug" will be hidden all over the city. More information Adding us as a Preferred Source in Google by using this link indicates that you would like to see more of our content in Google News results. Breakthroughs, discoveries, and DIY tips sent six days a week. A new gold rush is coming to California. For the third year, San Francisco's Witter Coin will host a treasure hunt across the city collectively worth over $50,000.

Studying these nematodes will help scientists plan for a long-term human presence on the moon. More information Adding us as a Preferred Source in Google by using this link indicates that you would like to see more of our content in Google News results. Breakthroughs, discoveries, and DIY tips sent six days a week. The Artemis II astronauts were back on Earth for less than a day before worms took their place in space. The space worms launched from Cape Canaveral, Florida, on April 11, aboard NASA's Commercial Resupply Services 24 mission (CRS-24) and are on a journey to the International Space Station (ISS).

Environment Animals Wildlife Birds Jackie and Shadow's 2026 babies: Everything you need to know about the new eaglets The two new chicks hatched in early April and are eating lots of fish, sleeping, and acting like siblings. More information Adding us as a Preferred Source in Google by using this link indicates that you would like to see more of our content in Google News results. The chicks are currently figuring out their "pecking order," or who gets first dibs on food. Breakthroughs, discoveries, and DIY tips sent six days a week. It's been another roller coaster nesting season for Jackie and Shadow, a pair of internet-famous bald eagle parents living in San Bernardino National Forest in Southern California.

Critical information about this unknown boy was destroyed during World War II. More information Adding us as a Preferred Source in Google by using this link indicates that you would like to see more of our content in Google News results. CT scanning and X-ray imaging allowed archaeologists to examine the mummy in extreme detail. Breakthroughs, discoveries, and DIY tips sent six days a week. Archaeologists in Poland are finally solving an over 2,000-year-old mummy mystery.

Insiders claim failed AI rollout could be to blame for Tim Cook's departure from Apple - as one says'the AI era requires a different kind of leadership' Australia has spoken: Report reveals what everyone is thinking about Prince Harry and Meghan Markle's Australia tour US troops board second tanker as Trump accuses Iran of violating ceasefire'numerous times' - Live updates New'Hollywood dose' pill: A-listers hooked on'youth elixir' that dermatologists say is anti-ageing, shrinks pores, smooths wrinkles... and even banishes rosacea Days after we got engaged, the love of my life told me he'd killed a man and buried him in a bog. I reported him to police... but then I made this irreversible mistake Papa John's under fire for an outrageous message now printed on all pizza boxes Fury as murderer marries pen pal behind bars... as teenage victim's mom says: 'I'm serving a life sentence without my son' Ritzy Bay Area town torn apart after teacher's daughter, 16, killed four friends in high-speed crash... then she posted a TikTok video that poured fuel on the flames Trump confronts Xi as US forces seize Chinese ship carrying mysterious'gift' to Iran How to lose weight when perimenopause sabotages your metabolism: I'm a PT but when I hit 46, I piled on the pounds overnight. New Jersey man's chilling'cancer map' fuels fears of poisoned neighborhood with 41 cases and counting Supreme Court secrets spill out: Insider names'hard a**' Justice who is'emotionally abusive' and leaves clerks with'fear in their eyes' AMANDA PLATELL: Why desperate Fergie's next move will be her biggest bombshell yet... and this is the only thing that can stop her I was losing hair so fast a bald spot the size of an orange appeared. I owe my life to a $1 at-home treatment that REVERSED the damage in a month. Even Cameron Diaz admits she's a dirty mess.