In this paper, we study Nyström subsampling for vector-valued regression in vector-valued reproducing kernel Hilbert spaces. Standard kernel methods often suffer from prohibitive computational costs due to the construction and inversion of large kernel matrices, which limits their scalability to large datasets. To overcome this bottleneck, we propose an efficient operator learning algorithm based on Nyström subsampling that accommodates functional outputs. Under general source conditions characterized by index functions-extending beyond the classical Hölder-type and operator-monotone frameworks-we establish minimax-optimal convergence rates for the proposed estimator. As an application of the proposed framework, we consider function denoising problems. Unlike classical denoising methods, which are typically tailored to specific signal representations or noise models, our approach formulates denoising within a general operator learning framework. Numerical experiments on signal denoising, real-time audio denoising, image denoising, inverse Radon transform reconstruction, and energy-efficiency prediction confirm that the proposed method achieves performance comparable to full kernel methods while substantially reducing computational cost.

This paper investigates convergence properties of regularized Nystr om subsampling applied to the unsupervised domain adaptation problem under covariate shift. We focus on the low-smoothness (misspecified) case where the target function lies outside the reproducing kernel Hilbert space. By combining Tikhonov regularization with Nystr om projection onto a subsampled subspace, we obtain upper bounds on the excess risk that hold with high probability and are expressed in terms of the source condition, the effective dimension, and the sample sizes. We further extend the analysis to the setting where the Radon-Nikodym derivative between the target and source marginal distributions is unknown and must be approximated, and we identify the minimal additional sample sizes required to maintain the same convergence rate as in the oracle case.

Instrumental variable (IV) analyses generally start by posing a structural equation: Y = hstructural(X)+ϵ, (1) where hstructural represents the causal effect of X on Y, and X and ϵ may be endogenous (E[ϵ | X] = 0). Then given an exogenous instrument Z satisfying the exclusion restriction, the common statistical solution given joint observations of W = (X,Y,Z) P is to conduct inference on some continuous linear functional h 7 EP[m(W;h)] of a solution h H to the linear equation implied by exclusion: TPh = rP, (2) where TP: H G maps h 7 argming GEP(h(X) g(Z))2, rP = argminr GEP(Y r(Z))2, and H, G are closed linear subspaces of square-integrable functions of X and of Z, respectively. For example, if these are all square-integrable functions, then (TPh)(Z) = EP[h(X) | Z] is the conditional expectation.

In this case, we view the geometric domain as the discrete 1d gridΩ = [1,...,d], and consider geometric transformationsG as subsets of the symmetric group of permutations ofd

Blind inverse problems arise in many experimental settings where the forward operator is partially or entirely unknown. In this context, methods developed for the non-blind case cannot be adapted in a straightforward manner. Recently, data-driven approaches have been proposed to address blind inverse problems, demonstrating strong empirical performance and adaptability. However, these methods often lack interpretability and are not supported by rigorous theoretical guarantees, limiting their reliability in applied domains such as imaging inverse problems. In this work, we shed light on learning in blind inverse problems within the simplified yet insightful framework of Linear Minimum Mean Square Estimators (LMMSEs). We provide an in-depth theoretical analysis, deriving closed-form expressions for optimal estimators and extending classical results. In particular, we establish equivalences with suitably chosen Tikhonov-regularized formulations, where the regularization depends explicitly on the distributions of the unknown signal, the noise, and the random forward operators. We also prove convergence results under appropriate source condition assumptions. Furthermore, we derive rigorous finite-sample error bounds that characterize the performance of learned estimators as a function of the noise level, problem conditioning, and number of available samples. These bounds explicitly quantify the impact of operator randomness and reveal the associated convergence rates as this randomness vanishes. Finally, we validate our theoretical findings through illustrative numerical experiments that confirm the predicted convergence behavior.

In 2021, OpenAI shocked the world by improving the zero-shot classification accuracy on ImageNet from 11.5% to 76.2% via the CLIP series of models (Radford et al., 2021). This event redefined the goal of zero-shot prediction from producing models that generalized to unseen classes to those that generalized to unseen tasks entirely. Two fundamental drivers of CLIP's success were 1) the use of natural language as a medium for representing arbitrary classes (as in the previous state-of-the-art Visual N-grams (Li et al., 2017)), and 2) a massive, yet carefully designed pre-training set which significantly impacted downstream performance Radford et al. (2021); Fang et al. (2023); Xu et al. (2024). Despite the remarkable success of these foundation model-based pipelines Bommasani et al. (2022), there are unique components of zero-shot prediction that warrant investigation from a theoretical point of view. To clarify these gaps, we contrast zero-shot prediction (ZSP) with the related setting of few-shot learning (FSL). Let x X denote an input (often an image) that accompanies a discrete value y Y (often a class label).

Artificial Intelligence has revolutionised critical care for common conditions. Yet, rare conditions in the intensive care unit (ICU), including recognised rare diseases and low-prevalence conditions in the ICU, remain underserved due to data scarcity and intra-condition heterogeneity. To bridge such gaps, we developed KnowRare, a domain adaptation-based deep learning framework for predicting clinical outcomes for rare conditions in the ICU. KnowRare mitigates data scarcity by initially learning condition-agnostic representations from diverse electronic health records through self-supervised pre-training. It addresses intra-condition heterogeneity by selectively adapting knowledge from clinically similar conditions with a developed condition knowledge graph. Evaluated on two ICU datasets across five clinical prediction tasks (90-day mortality, 30-day readmission, ICU mortality, remaining length of stay, and phenotyping), KnowRare consistently outperformed existing state-of-the-art models. Additionally, KnowRare demonstrated superior predictive performance compared to established ICU scoring systems, including APACHE IV and IV-a. Case studies further demonstrated KnowRare's flexibility in adapting its parameters to accommodate dataset-specific and task-specific characteristics, its generalisation to common conditions under limited data scenarios, and its rationality in selecting source conditions. These findings highlight KnowRare's potential as a robust and practical solution for supporting clinical decision-making and improving care for rare conditions in the ICU.