The natural gradient method is widely used in statistical optimization, but its standard formulation assumes a Euclidean parameter space. This paper proposes an inversion-free stochastic natural gradient method for probability distributions whose parameters lie on a Riemannian manifold. The manifold setting offers several advantages: one can implicitly enforce parameter constraints such as positive definiteness and orthogonality, ensure parameters are identifiable, or guarantee regularity properties of the objective like geodesic convexity. Building on an intrinsic formulation of the Fisher information matrix (FIM) on a manifold, our method maintains an online approximation of the inverse FIM, which is efficiently updated at quadratic cost using score vectors sampled at successive iterates. In the Riemannian setting, these score vectors belong to different tangent spaces and must be combined using transport operations. We prove almost-sure convergence rates of $O(\log{s}/s^α)$ for the squared distance to the minimizer when the step size exponent $α>2/3$. We also establish almost-sure rates for the approximate FIM, which now accumulates transport-based errors. A limited-memory variant of the algorithm with sub-quadratic storage complexity is proposed. Finally, we demonstrate the effectiveness of our method relative to its Euclidean counterparts on variational Bayes with Gaussian approximations and normalizing flows.

Sydney Sweeney's role is cut from The Devil Wears Prada 2 Driver who hit and killed jogger father-of-two sues victim's estate claiming incident left him with severe PTSD New'Hollywood dose' pill: A-listers hooked on'youth elixir' that dermatologists say is anti-aging, shrinks pores, smooths wrinkles... and even banishes rosacea Alarm over popular new coffee chain invading the US... as experts warn of chilling secret behind its $1.99 brew Vance grounded at White House as Iran peace talks in turmoil and Trump declares: 'I expect to be bombing' Jordon Hudson extends her control over Bill Belichick's empire with secret move that is set to leave his family and friends furious Ark of the Covenant's final resting place pinpointed by archaeologists as fresh search begins Life-threatening cantaloupe recall in four states upgraded to FDA's highest risk level... 'reasonable probability of death' Truth about your Mounjaro injection site: Our expert doctors reveal exactly where you should inject yourself for the best results, what to do if your weight loss has slowed down... and the areas you should NEVER jab Ritzy Bay Area town torn apart after teacher's daughter, 16, crashed car while speeding and killed four friends... then posted a TikTok video that poured fuel on the flames Beloved Republican mayor of small Great Plains town could be deported over'mistake' he insists was an innocent one Humiliating moment runner celebrates winning marathon... only to be pipped at the line by rival in brutal finish The new'posh' drug that's easier to order than Uber Eats - and why all my middle-class friends have ditched booze and cocaine for it: JANA HOCKING Why desperate Fergie's next move will be her biggest bombshell yet... and this is the only thing that can stop her: AMANDA PLATELL RED MORE: Man's best friend has been in Britain for 14,300 years Humans began gambling 12,000 years ago, experts say - after discovering dice that date back to the last Ice Age. A team from Colorado State University have unearthed the earliest evidence of two-sided dice crafted from small pieces of bone. They were originally found at an archaeological site on the western Great Plains of America, predating the current oldest known dice by more than 6,000 years. The discovery indicates that gambling and games of chance have been a persistent feature of North American culture since the end of the last Ice Age, experts say. 'Historians have traditionally treated dice and probability as Old World innovations,' researcher Robert Madden said.

The poster child of the AI boom, valued at $850bn, needs to show strategic discipline after'casting its net too wide' If OpenAI is going to float this year, it has to get serious about its business model. The wow factor around the US company - the poster child of an AI industry boom that has stoked fears of a stock market bubble - has been long established, but when will the profits come? The developer of ChatGPT is one of the biggest startups in the world and is now valued at $850bn (£645bn). Meanwhile, it is reportedly spending $600bn on infrastructure (the amount it invests in datacentres and chips to power its AI models) by 2030. At least this is a reduction on an initial estimate of $1.4tn .

Parametric models deployed in non-stationary environments degrade as the underlying data distribution evolves over time (a phenomenon known as temporal domain drift). In the current work, we present KOMET (Koopman Operator identification of Model parameter Evolution under Temporal drift), a model-agnostic, data-driven framework that treats the sequence of trained parameter vectors as the trajectory of a nonlinear dynamical system and identifies its governing linear operator via Extended Dynamic Mode Decomposition (EDMD). A warm-start sequential training protocol enforces parameter-trajectory smoothness, and a Fourier-augmented observable dictionary exploits the periodic structure inherent in many real-world distribution drifts. Once identified, KOMET's Koopman operator predicts future parameter trajectories autonomously, without access to future labeled data, enabling zero-retraining adaptation at deployment. Evaluated on six datasets spanning rotating, oscillating, and expanding distribution geometries, KOMET achieves mean autonomous-rollout accuracies between 0.981 and 1.000 over 100 held-out time steps. Spectral and coupling analyses further reveal interpretable dynamical structure consistent with the geometry of the drifting decision boundary.

Online health communities (OHCs) are vital for fostering peer support and improving health outcomes. Support groups within these platforms can provide more personalized and cohesive peer support, yet traditional support group formation methods face challenges related to scalability, static categorization, and insufficient personalization. To overcome these limitations, we propose two novel machine learning models for automated support group formation: the Group specific Dirichlet Multinomial Regression (gDMR) and the Group specific Structured Topic Model (gSTM). These models integrate user generated textual content, demographic profiles, and interaction data represented through node embeddings derived from user networks to systematically automate personalized, semantically coherent support group formation. We evaluate the models on a large scale dataset from MedHelp, comprising over 2 million user posts. Both models substantially outperform baseline methods including LDA, DMR, and STM in predictive accuracy (held out log likelihood), semantic coherence (UMass metric), and internal group consistency. The gDMR model yields group covariates that facilitate practical implementation by leveraging relational patterns from network structures and demographic data. In contrast, gSTM emphasizes sparsity constraints to generate more distinct and thematically specific groups. Qualitative analysis further validates the alignment between model generated groups and manually coded themes, showing the practical relevance of the models in informing groups that address diverse health concerns such as chronic illness management, diagnostic uncertainty, and mental health. By reducing reliance on manual curation, these frameworks provide scalable solutions that enhance peer interactions within OHCs, with implications for patient engagement, community resilience, and health outcomes.

Many causal discovery algorithms, including the celebrated FCI algorithm, output a Partial Ancestral Graph (PAG). PAGs serve as an abstract graphical representation of the underlying causal structure, modeled by directed acyclic graphs with latent and selection variables. This paper develops a characterization of the set of extended-type conditional independence relations that are invariant across all causal models represented by a PAG. This theory allows us to formulate a general measure-theoretic version of Pearl's causal calculus and a sound and complete identification algorithm for PAGs under selection bias. Our results also apply when PAGs are learned by certain algorithms that integrate observational data with experimental data and incorporate background knowledge.

The theory of Local Intrinsic Dimensionality (LID) has become a valuable tool for characterizing local complexity within and across data manifolds, supporting a range of data mining and machine learning tasks. Accurate LID estimation requires samples drawn from small neighborhoods around each query to avoid biases from nonlocal effects and potential manifold mixing, yet limited data within such neighborhoods tends to cause high estimation variance. As a variance reduction strategy, we propose an ensemble approach that uses subbagging to preserve the local distribution of nearest neighbor (NN) distances. The main challenge is that the uniform reduction in total sample size within each subsample increases the proximity threshold for finding a fixed number k of NNs around the query. As a result, in the specific context of LID estimation, the sampling rate has an additional, complex interplay with the neighborhood size, where both combined determine the sample size as well as the locality and resolution considered for estimation. We analyze both theoretically and experimentally how the choice of the sampling rate and the k-NN size used for LID estimation, alongside the ensemble size, affects performance, enabling informed prior selection of these hyper-parameters depending on application-based preferences. Our results indicate that within broad and well-characterized regions of the hyper-parameters space, using a bagged estimator will most often significantly reduce variance as well as the mean squared error when compared to the corresponding non-bagged baseline, with controllable impact on bias. We additionally propose and evaluate different ways of combining bagging with neighborhood smoothing for substantial further improvements on LID estimation performance.

We propose stepwise variational inference (VI) with vine copulas: a universal VI procedure that combines vine copulas with a novel stepwise estimation procedure of the variational parameters. Vine copulas consist of a nested sequence of trees built from copulas, where more complex latent dependence can be modeled with increasing number of trees. We propose to estimate the vine copula approximate posterior in a stepwise fashion, tree by tree along the vine structure. Further, we show that the usual backward Kullback-Leibler divergence cannot recover the correct parameters in the vine copula model, thus the evidence lower bound is defined based on the Rényi divergence. Finally, an intuitive stopping criterion for adding further trees to the vine eliminates the need to pre-define a complexity parameter of the variational distribution, as required for most other approaches. Thus, our method interpolates between mean-field VI (MFVI) and full latent dependence. In many applications, in particular sparse Gaussian processes, our method is parsimonious with parameters, while outperforming MFVI.

Tech firm'says goodbye' to Sora, made publicly available in 2024, just six months after its launch of a stand-alone app In an abrupt announcement on Tuesday, OpenAI said it was "saying goodbye" to its AI video generator Sora. The move comes just six months after the company's splashy launch of a stand-alone app with which people could make and share hyper-realistic AI videos in a scrolling social feed. "To everyone who created with Sora, shared it, and built community around it: thank you," the company wrote in a post on X . "What you made with Sora mattered, and we know this news is disappointing." OpenAI first made Sora publicly available in late 2024, but it wasn't until the company launched Sora 2 and its stand-alone app last September that the video generator reached mainstream attention.

Panel data methods are widely used in empirical analysis to address unobserved heterogeneity, but causal inference remains challenging when treatments are endogenous and confounding variables high-dimensional and potentially nonlinear. Standard instrumental variables (IV) estimators, such as two-stage least squares (2SLS), become unreliable when instrument validity requires flexibly conditioning on many covariates with potentially non-linear effects. This paper develops a Double Machine Learning estimator for static panel models with endogenous treatments (panel IV DML), and introduces weak-identification diagnostics for it. We revisit three influential migration studies that use shift-share instruments. In these settings, instrument validity depends on a rich covariate adjustment. In one application, panel IV DML strengthens the predictive power of the instrument and broadly confirms 2SLS results. In the other cases, flexible adjustment makes the instruments weak, leading to substantially more cautious causal inference than conventional 2SLS. Monte Carlo evidence supports these findings, showing that panel IV DML improves estimation accuracy under strong instruments and delivers more reliable inference under weak identification.