We study minimal attention-only transformers under all-token corruption and show they admit a two-stage empirical Bayes interpretation. A single attention step computes a kernel-weighted posterior mean with respect to the empirical distribution defined by the context. Depth refines this distribution through particle dynamics (Stage 1), while a long-range skip-connection carries the noisy input as a query for posterior inference (Stage 2), revealing distinct statistical roles for depth and attention residuals. The framework isolates a minimal setting in which the context itself induces a depth-dependent energy landscape governing in-context inference. We show that effective denoising can emerge without an explicit noise schedule: a fixed kernel bandwidth and finite integration horizon suffice, yielding a principled depth-noise relationship. We further establish a posterior-mean recovery guarantee for a class of well-behaved priors, where the empirical estimator converges to the Bayes-optimal predictor under asymptotic conditions. Connecting these dynamics to reverse-diffusion limits, our results provide a statistical interpretation of attention as in-context inference via sample-based posterior estimation, without explicit density modeling.

Sparse recovery in linear systems underpins applications from signal processing to high-dimensional regression. Sparse Bayesian Learning, grounded in the principle of automatic relevance determination (ARD), offers a practical Bayesian mechanism for feature sparsity via marginal likelihood optimization. Yet, its reliance on a homoscedastic noise model renders it sensitive to data contaminations such as outliers or misspecified noise, harming model fit and predictions. Instead, we propose jointly learning individual feature and sample relevancies, enabling simultaneous model and data sparsification via a single Bayesian objective. This symmetric pruning of model and data offers a natural extension that preserves conjugacy, admits closed-form updates for standard optimization procedures, and aligns with perspectives from robust regression and influence functions. Empirical results across diverse regression tasks affirm that a joint ARD approach consistently yields both sparse and robust prediction models.

Despite strong average-case performance, deep learning models often exhibit systematic errors on specific population groups, known as error slices. Identifying these groups and the root causes of their failures is critical for model debugging and bias mitigation. However, existing error Slice Discovery Methods (SDMs) typically generate explanations disconnected from the model's inference process, thus only approximating the underlying error source and may be inaccurate. We address this limitation by leveraging Concept Bottleneck Models (CBMs), whose predictions are directly dependent on human-understandable semantic concepts. Since downstream task failures in CBMs commonly arise from concept mispredictions, concept representations provide a strong candidate for error slice identification, offering fine-grained explanations directly linked to the error source. Building on this insight, we introduce CB-SLICE, a concept-based SDM that groups samples with shared concept prediction failures and identifies the keyword concepts most responsible for each slice's failure mode. Across multiple benchmarks, we show that CB-SLICE outperforms state-of-the-art methods in uncovering well-known biases while providing richer and more faithful explanations of model errors.

Modern learning systems excel at interpolation but struggle to generalize to unseen tasks outside the training distribution's support. This failure occurs even in simple settings, such as handling task parameters beyond the training range, and persists despite advances in foundation models. To this end, we develop the Relational Task Extrapolator (RTE), an algorithm designed to enable systematic extrapolation to novel tasks. The key observation is that extrapolation is inherently relational: extrapolating to unseen tasks requires learning how tasks transform into one another. If a model learns the transformation between tasks A and B during training, it can apply that same transformation to relate known tasks to unseen ones at test time. RTE operationalizes this idea by decomposing each target task into a known anchor task and a transformation linking the anchor and target. It then learns a relational operator, mapping an anchor-transformation pair to predictions for the target task. We instantiate RTE across multiple task extrapolation regimes in function prediction, e.g. where target tasks use out-of-range parameters (parameter extrapolation), have greater compositional depth (length extrapolation), and/or recombine function primitives in unseen ways (compositional extrapolation). We further extend RTE to sequence prediction, integrating it into fine-tuning algorithms for foundation models. Across empirical studies, we find that RTE substantially outperforms existing approaches on extrapolation to novel, unseen tasks.

Score-based diffusion models have demonstrated remarkable empirical success in learning high-dimensional distributions, particularly those exhibiting low-dimensional and multi-modal structures. However, theoretical understanding of their statistical efficiency remains limited. Existing theories typically rely on strong regularity assumptions, such as uniformly bounded densities or globally smooth score functions, which fail to capture such intrinsic structures. In this work, we study the sample complexity of diffusion models for learning distributions supported on a union of low-dimensional subspaces. Assuming that the data distribution within each subspace is subgaussian, we show that diffusion models require at most $\widetilde{O}(\varepsilon^{-k \vee 2})$ samples to achieve $\varepsilon$ error in 1-Wasserstein distance, where $k$ is the intrinsic dimension. This near-optimal convergence rate depends only on the intrinsic dimension and significantly improves upon prior theoretical guarantees that suffer from the curse of dimensionality. Notably, our analysis applies to a broad collection of distributions without imposing smoothness, bounded-density, or log-concavity assumptions. Overall, our results show that diffusion models can statistically adapt to intrinsic low-dimensional structure while naturally accommodating multi-modal data, offering a rigorous theoretical justification for their success in complex high-dimensional learning tasks.

Predicting a complete spatially correlated field from sparse observations is a fundamental challenge in spatial statistics and environmental modelling. Classical interpolation methods such as Kriging rely on Gaussian process assumptions and variography, which can limit their effectiveness in non-stationary settings and require substantial domain expertise. In this work, we leverage an architecture based on convolutional neural networks (CNNs) for spatial interpolation that is trained and applied on a single partially observed field, without access to external data or prior fields. The model is supervised directly on the observed locations and learns to predict values at unobserved points on the user defined grid. Unlike Kriging, our method does not require explicit covariance modelling or variogram estimation, and it can flexibly capture local spatial patterns in a data-driven manner. This work demonstrates the potential of CNNs for single-instance spatial interpolation under sparse supervision, offering a practical alternative to classical geostatistical methods, and extending the use of CNNs to a new problem domain.

Cluster analysis is a widely applied machine learning technique to understand the existing patterns in the population of gamma-ray bursts (GRBs), in order to explore their physical sources. In the present scenario, the number of clusters corresponding to differentiable groups is still under conflict, in spite of numerous attempts with the state-of-the-art clustering procedures. This crucial unknown parameter needs to be evaluated, either directly or indirectly in terms of other tuning parameters, to produce the clusters in GRBs through implementation of an appropriate clustering algorithm. While most of the applied algorithms reached two physically explained groups of merger and collapsar predominated by the short and long bursts respectively, other statistical approaches violated this binary partition. However, physical establishment of any additional cluster(s) is not yet confirmed. Therefore, we propose a new algorithm, from a different stream of clustering referred to as `completely parameter-free', which carries out the classification of GRBs in a manner that has not been tried so far. It indicates two main groups, of short and long duration bursts from the BATSE sample, compatible with the merger-collapsar theory.

Reliable probability estimates are critical in many machine learning applications, yet modern classifiers are often poorly calibrated. Post-hoc calibration provides a simple and widely used solution, but the large number of proposed methods, combined with small-scale and inconsistent evaluations, makes it difficult to determine which approaches are truly effective in practice. We introduce a large-scale, standardized benchmark for post-hoc calibration, covering nearly 2000 experiments across tabular and computer vision tasks, including binary, multiclass, and large-scale classification settings. Our benchmark aggregates predictions from a diverse set of classical models, modern deep learning architectures, and foundation models, and provides unified, reproducible implementations of dozens of calibration methods within a common evaluation framework. We argue that Post-Hoc Improvement (PHI) in proper scoring rules offers a principled alternative to traditional calibration error estimators for comparing post-hoc methods, capturing both calibration quality and potential degradation to the model's predictive performance. Using this framework, we conduct the most comprehensive empirical study of post-hoc calibration to date. Our results reveal consistent patterns across domains: smooth calibration functions outperform binning-based approaches, dedicated multiclass methods are essential in high-dimensional settings, and generic machine learning models are not competitive without calibration-specific design. To facilitate future research, we release all data, code, and evaluation tools, providing a plug-and-play benchmark for developing and comparing calibration methods.

Finding approximations to an intractable probability distribution π of interest (usually known only up to a normalizing constant) is a key problem in scientific computing. Variational Inference stands out as a particularly attractive tool for this task, owing to its statistical and computational efficiency, and it has been the framework underlying many advances in computational statistics over the past half century (Parisi, 1980; Hinton and Van Camp, 1993; Jordan et al., 1999; Bishop and Nasrabadi, 2006). The central idea is to seek a tractable approximation to π within a chosen family of tractable distributions Q by minimizing a divergence to π over that'variational' family. Often, it is convenient or well-motivated to work with the family of product (or tensor, or factorized) distributions Q = P m, and define optimality through minimisation of the Kullback-Leibler (KL) divergence (also'relative entropy') min KL(ϱ||π): ϱ P m . A key practical aspect of working with this particular loss function is that in solving the associated optimisation problem, one is only required to compute expectations under the tractable variational distribution ϱ, rather than under the intractable target distribution π. In Bayesian statistics, π typically represents the joint posterior distribution of latent variables z Z and some parameters β B given observed data y Y. In these cases, we often choose m = 2 and seek the best variational approximation µ(dz) ν(dβ) to π to solve min KL(µ ν||π): µ P(Z), ν P(B) . The coordinate ascent variational inference algorithm (CAVI, Bishop and Nasrabadi, 2006; Blei et al., 2017) solves this problem by iteratively minimizing the Kullback-Leibler divergence with respect to one element at a time: given a starting point ν0, it iterates µk:= argmin

Numeric tabular datasets are the dominant data format in scientific practice, yet large language models lack native mechanisms for representing numeric datasets in a meaningful way across heterogeneous feature spaces. Existing approaches either target predictive modeling over individual datasets, which requires a shared set of variable definitions, or lack mechanisms for interpretable cross-dataset alignment. The proposed methodology characterizes numeric tabular datasets through structured exploratory data analysis descriptors, embeds those descriptors into a shared vector space using a pretrained sentence transformer, and quantifies cross-dataset similarity via Canonical Correlation Analysis (CCA). Furthermore, a penalized formulation of CCA is applied to recover sparse, interpretable variable-level correspondences between datasets, identifying which statistical descriptors or variable-level quantities drive cross-dataset alignment without requiring shared variable names or feature conventions. Differential privacy is optionally applied to the descriptor set prior to embedding, supporting deployment in sensitive data contexts without requiring access to raw observations at time of comparison. The methodology is evaluated across 15 datasets spanning general-purpose benchmarks, materials informatics, and nuclear-grade graphite characterization. Results demonstrate a total P@1 score of 0.9, with known nearest-neighbor retrieval and cluster structure remaining robust across embedding ablations and differential privacy budgets. The proposed framework provides a principled pathway for integrating heterogeneous numeric data into retrieval-augmented generation pipelines while preserving statistical context, with direct applications to data-driven algorithm selection and simulation model initialization for unknown datasets.