We introduce a simple yet powerful framework for training large language models. In contrast to the standard autoregressive next-token prediction based on an exact prefix, we propose a perturbation-based procedure that first transforms the prefix into a semantic neighbor and then conditions on this perturbed variant for next-token prediction. This yields a hierarchical model with a pre-post-additive noise structure. Within this framework, we develop a rigorous theory of extrapolability, namely, the capacity of a model class to make reliable predictions for token sequences that lie outside the empirical support of the training corpus. We evaluate the finite-sample performance of the proposed procedure using both synthetic and real-world language data. Results show that the proposed method consistently improves out-of-support prediction while maintaining competitive in-support performance, demonstrating that perturbation offers a practical route to language modeling.

Causal discovery methods such as LiNGAM identify causal structure from observational data by assuming mutually independent disturbances. This assumption is fragile: shared volatility, common scale effects, or other forms of dependence can cause the methods to recover the wrong causal order, even with infinite data. We introduce the Linear Mean-Independent Acyclic Model (LiMIAM), which replaces full independence with weaker one-sided mean-independence restrictions on the disturbances. Under finite-order consequences of these restrictions, source nodes are generically identifiable, and hence a compatible causal order can be recovered recursively. Our proof is constructive and leads to DirectLiMIAM, a sequential residual-based algorithm for causal discovery under dependent noise. In simulations with mean-independent but dependent disturbances, DirectLiMIAM outperforms LiNGAM methods. A large-scale empirical application to the oil market highlights the implausibility of the independence assumption and the ability of DirectLiMIAM to recover a realistic causal ordering, from policy to production and from prices to inflation.

Large-scale population-level datasets, such as the UK Biobank and the All of Us Research Program, often lack covariates needed for a specific analysis, such as genetic or lifestyle measures, while related studies measure them. This creates a cross-population missing data problem in which covariates are completely unobserved in the target population, rather than partially missing within one dataset. We propose an augmented transfer regression learning method for this setting. The key identifying condition is a sub-population shift assumption: the joint distribution of the outcome and observed covariates may differ across source and target populations, but the conditional distribution of the missing covariates given observed variables is invariant. We combine importance-weighted estimating equations with imputation terms for first- and second-order moments of the missing covariates. The resulting estimator is doubly robust, remaining consistent if either the density ratio model or both imputation models are correctly specified. It is $n^{1/2}$-consistent and asymptotically normal, and attains the semiparametric efficiency bound when both nuisance models are correctly specified.

Single-cell ATAC-seq (scATAC-seq) enables high-resolution mapping of chromatin accessibility, yet privacy regulations and data size constraints hinder multi-institutional sharing. Federated learning (FL) offers a privacy-preserving alternative, but faces three fundamental barriers in scATAC-seq analysis: ultra-high dimensionality, extreme sparsity, and severe cross-institutional heterogeneity. We propose FL-Sailer, the first FL framework designed for scATAC-seq data. FL-Sailer integrates two key innovations: (i) adaptive leverage score sampling, which selects biologically interpretable features while reducing dimensionality by 80%, and (ii) an invariant VAE architecture, which disentangles biological signals from technical confounders via mutual information minimization. We provide a convergence guarantee, showing that FL-Sailer converges to an approximate solution of the original high-dimensional problem with bounded error. Extensive experiments on synthetic and real epigenomic datasets demonstrate that FL-Sailer not only enables previously infeasible multi-institutional collaborations but also surpasses centralized methods by leveraging adaptive sampling as an implicit regularizer to suppress technical noise. Our work establishes that federated learning, when tailored to domain-specific challenges, can become a superior paradigm for collaborative epigenomic research.

Inferring continuum models directly from video is hampered by two facts: the recorded field is uncalibrated image intensity rather than a physical state, and direct numerical differentiation of noisy frames is unstable. We develop a video-to-PDE pipeline that converts grayscale recordings of an ink plume into a normalised scalar field $u(x,y,t)$, isolates a bulk drift $\mathbf{v}(t)$ from intrinsic spreading via the intensity-weighted centroid, and identifies an effective transport law by weak-form sparse regression. Conditioning, threshold-sweep and random-centre diagnostics show that overcomplete libraries are strongly collinear; the search is therefore restricted to compact gradient-based libraries. Coefficients are refined by an inverse physics-informed network and recalibrated against forward rollouts, with a chronological block bootstrap quantifying uncertainty. The selected reduced model $u_t+\mathbf v(t)\!\cdot\!\nabla u = 9.005\,|\nabla u|^{2}+0.666\,Δu$ outperforms advection--diffusion baselines on held-out frames, retains a positive Laplacian coefficient, and admits a Cole--Hopf reduction to a linear advection--diffusion equation. The framework demonstrates that uncalibrated visual data can yield compact, predictive and structurally interpretable continuum models when discovery, calibration and uncertainty are treated as distinct stages.

Large language models hallucinate in predictable ways: attention routing fails by over-concentrating on a narrow set of positions, or by spreading so diffusely that relevance is diluted, and the shape of the failure carries diagnostic signal. A widely used family of spectral methods analyzes the symmetric component of the degree-normalized attention operator, which governs transport capacity; we prove that every transpose-invariant spectral diagnostic of this operator is structurally orientation-blind (it cannot distinguish an operator from its transpose, and therefore cannot detect information-flow direction), with a quantitative converse establishing the asymmetry coefficient $G$ as the unique control parameter for direction. Pairing this with a closed-form bipartite-Cheeger landscape for canonical causal architectures, we show that uniform causal attention satisfies an $n$-independent floor $ϕ\ge 1/5$ with worst cut at $t^\ast/n \approx 0.32$, while window attention pierces the floor as $O(w/n)$; failure modes are shape-different, not just value-different. The resulting two-axis diagnostic ($ϕ$ for capacity, $G$ for direction) yields a falsifiable polarity prediction: bottleneck- and diffuse-dominated benchmarks should exhibit opposite polarity. Under length-controlled evaluation, transport features retain interpretable signal (LC-AUROC from 0.62 to 0.84) on tested models up to 8B parameters, with polarity reversing as predicted between HaluEval and MedHallu.

Published transfer-BO comparisons often estimate an average treatment effect of acquisition choice over hidden regime variables, while practitioners need the conditional effect for their specific prior quality, budget ratio, and metric. An audit of 40 transfer-BO papers from NeurIPS, ICML, ICLR, AISTATS, UAI, TMLR, JMLR, and AutoML-Conf (2022-2025) finds that 98% never vary B/|A| as a controlled axis. On the same GDSC2 benchmark, changing only the budget reverses the ranking: at B=50, Greedy outperforms UCB by 0.050 Hit@1, while at B=100, UCB outperforms Greedy by 0.035. We capture this transition with the Portable Regime Score PRS=(B/|A|)(1-rho), where rho is the prior rank correlation and can be estimated from pilot contexts before the main comparison. Across 79 conditions spanning chemistry, drug-response biology, and HPO, a hierarchical model gives beta=0.50 (p=1.1e-9), and 19% of conditions fall in an equivalence zone where |advantage|<0.01 Hit@1. In five published reversal cases, PRS predicts the winner from pre-comparison observables. A No-Free-Leaderboard proposition explains why unconditional rankings are unstable: when CATE changes sign across regimes, the reported ATE becomes a function of benchmark mixture. RegimePlanner, which estimates rho online and switches acquisition accordingly, wins all 16 HPO-B search spaces at B=100 and exceeds the matched {Greedy,UCB} per-context oracle on GDSC2 by 18%. Pre-registered predictions achieve 27/40=67.5% overall accuracy and above 90% within EMA prior families. The practical protocol is simple: report B/|A|, rho, K, and metric alongside any claimed acquisition advantage.

Despite theoretical advantages, causal methods for Gene Regulatory Network (GRN) inference from single-cell RNA-seq data consistently fail to match or outperform correlation-based baselines in many realistic benchmarks, a persistent puzzle which casts doubt on the value of causality for this task. We argue that existing benchmarks are insufficiently controlled to answer this question because they evaluate on real or semi-real data where multiple pathologies co-occur, confounding failure modes, and obscuring the specific conditions under which different inference methods excel or fail. To address this gap, we introduce a controlled diagnostic framework that isolates seven biologically motivated pathologies (dropout, latent confounders, cell-type mixing, feedback loops, network density, sample size, and pseudotime drift) and measure how six representative methods spanning three inference paradigms degrade as each pathology intensifies. Across 6,120 controlled experiments, we find that causal methods genuinely dominate in clean and structurally favorable regimes, but specific pathologies (notably dropout and latent confounders) selectively neutralize their advantages. We further introduce an errortype decomposition that reveals methods with similar aggregate accuracy commit qualitatively different errors. To probe whether single-pathology effects persist when multiple stressors co-occur, we perform an interaction sweep over the three most impactful pathologies and find that their joint effects are sub-additive, while also exposing density-conditional cross-overs invisible to single-dial analysis. Our findings offer a nuanced understanding of when and why different methods succeed or fail for GRN inference, providing actionable insights for method development and practical guidance for practitioners.3

Hypergraphs model higher-order interactions, but realistic hypergraph generation remains difficult because incidence, hyperedge-size heterogeneity, and overlap structure are not faithfully captured by pairwise reductions. We propose \HEDGE, a generative model defined directly on relaxed incidence matrices via a structured stochastic diffusion. The forward process combines a hypergraph-specific two-sided heat operator with an Ornstein--Uhlenbeck component, preserving structure-aware noising near the data while yielding an explicit Gaussian terminal law. Conditional on an observed hypergraph, this forward process is linear-Gaussian, so conditional means, covariances, scores, and reverse-drift targets are available in closed form. We therefore learn a permutation-equivariant state-only reverse-drift field in incidence space by regressing onto exact conditional targets, and generate samples by simulating a learned reverse-time SDE from the Gaussian base law. We establish exactness in the ideal state-only setting together with finite-horizon stability guarantees, and empirically show improved hypergraph generation quality relative to strong baselines.

Regularization is often used in high-dimensional regression settings to generate a sparse model, which can save tremendous computing resources and identify predictors that are most strongly associated with the response. When the predictors can be represented by a Gaussian graphical model, the structure of the predictor graph can be exploited during regularization. Our proposed model exploits this underlying predictor graph structure by decomposing the estimated coefficient vector into a sum of latent variables that correspond to the sum of each node contribution to the coefficient vector. Regularization is then performed on the latent variables rather than on the coefficient vector directly. We use a penalty function that permits a clear user-defined trade-off between the L1 and L2 penalties and propose a novel proximal projection during optimization. Further, our implementation computes the projection operator for the intersection of selected groups, which conserves more computing resources compared to predictor duplication methods, especially for high-dimensional data. Through simulation, we evaluate the performance of our approach under different graph structures and node counts, and present results on real-world data. Results suggest that our method exhibits stable performance relative to other singly or doubly sparse graphical regression models.