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Learning tasks play an increasingly prominent role in quantum information and computation. They range from fundamental problems such as state discrimination and metrology over the framework of quantum probably approximately correct (PAC) learning, to the recently proposed shadow variants of state tomography. However, the many directions of quantum learning theory have so far evolved separately. We propose a general mathematical formalism for describing quantum learning by training on classical-quantum data and then testing how well the learned hypothesis generalizes to new data. In this framework, we prove bounds on the expected generalization error of a quantum learner in terms of classical and quantum information-theoretic quantities measuring how strongly the learner's hypothesis depends on the specific data seen during training. To achieve this, we use tools from quantum optimal transport and quantum concentration inequalities to establish non-commutative versions of decoupling lemmas that underlie recent information-theoretic generalization bounds for classical machine learning. Our framework encompasses and gives intuitively accessible generalization bounds for a variety of quantum learning scenarios such as quantum state discrimination, PAC learning quantum states, quantum parameter estimation, and quantumly PAC learning classical functions. Thereby, our work lays a foundation for a unifying quantum information-theoretic perspective on quantum learning.

We consider the general problem of Bayesian binary regression and we introduce a new class of distributions, the Perturbed Unified Skew Normal (pSUN, henceforth), which generalizes the Unified Skew-Normal (SUN) class. We show that the new class is conjugate to any binary regression model, provided that the link function may be expressed as a scale mixture of Gaussian densities. We discuss in detail the popular logit case, and we show that, when a logistic regression model is combined with a Gaussian prior, posterior summaries such as cumulants and normalizing constants can be easily obtained through the use of an importance sampling approach, opening the way to straightforward variable selection procedures. For more general priors, the proposed methodology is based on a simple Gibbs sampler algorithm. We also claim that, in the p > n case, the proposed methodology shows better performances - both in terms of mixing and accuracy - compared to the existing methods. We illustrate the performance through several simulation studies and two data analyses.

This paper considers the extension of data-enabled predictive control (DeePC) to nonlinear systems via general basis functions. Firstly, we formulate a basis functions DeePC behavioral predictor and we identify necessary and sufficient conditions for equivalence with a corresponding basis functions multi-step identified predictor. The derived conditions yield a dynamic regularization cost function that enables a well-posed (i.e., consistent) basis functions formulation of nonlinear DeePC. To optimize computational efficiency of basis functions DeePC we further develop two alternative formulations that use a simpler, sparse regularization cost function and ridge regression, respectively. Consistency implications for Koopman DeePC as well as several methods for constructing the basis functions representation are also indicated. The effectiveness of the developed consistent basis functions DeePC formulations is illustrated on a benchmark nonlinear pendulum state-space model, for both noise free and noisy data.

In this paper we construct new fully decoupled and high-order implicit-explicit (IMEX) schemes for the two-phase incompressible flows based on the new generalized scalar auxiliary variable approach with optimal energy approximation (EOP-GSAV) for Cahn-Hilliard equation and consistent splitting method for Navier-Stokes equation. These schemes are linear, fully decoupled, unconditionally energy stable, only require solving a sequence of elliptic equations with constant coefficients at each time step, and provide a new technique to preserve the consistency between original energy and modified energy. We derive that numerical solutions of these schemes are uniformly bounded without any restriction on time step size. Furthermore, we carry out a rigorous error analysis for the first-order scheme and establish optimal global error estimates for the phase function, velocity and pressure in two and three-dimensional cases. Numerical examples are presented to validate the proposed schemes.

Pretrained transformers exhibit the remarkable ability of in-context learning (ICL): they can learn tasks from just a few examples provided in the prompt without updating any weights. This raises a foundational question: can ICL solve fundamentally $\textit{new}$ tasks that are very different from those seen during pretraining? To probe this question, we examine ICL's performance on linear regression while varying the diversity of tasks in the pretraining dataset. We empirically demonstrate a $\textit{task diversity threshold}$ for the emergence of ICL. Below this threshold, the pretrained transformer cannot solve unseen regression tasks, instead behaving like a Bayesian estimator with the $\textit{non-diverse pretraining task distribution}$ as the prior. Beyond this threshold, the transformer significantly outperforms this estimator; its behavior aligns with that of ridge regression, corresponding to a Gaussian prior over $\textit{all tasks}$, including those not seen during pretraining. Thus, when pretrained on data with task diversity greater than the threshold, transformers $\textit{can}$ optimally solve fundamentally new tasks in-context. Importantly, this capability hinges on it deviating from the Bayes optimal estimator with the pretraining distribution as the prior. This study also explores the effect of regularization, model capacity and task structure and underscores, in a concrete example, the critical role of task diversity, alongside data and model scale, in the emergence of ICL. Code is available at //github.com/mansheej/icl-task-diversity.

Causal investigations in observational studies pose a great challenge in scientific research where randomized trials or intervention-based studies are not feasible. Leveraging Shannon's seminal work on information theory, we consider a framework of asymmetry where any causal link between putative cause and effect must be explained through a mechanism governing the cause as well as a generative process yielding an effect of the cause. Under weak assumptions, this framework enables the assessment of whether X is a stronger predictor of Y or vice-versa. Under stronger identifiability assumptions our framework is able to distinguish between cause and effect using observational data. We establish key statistical properties of this framework. Our proposed methodology relies on scalable non-parametric density estimation using fast Fourier transformation. The resulting estimation method is manyfold faster than the classical bandwidth-based density estimation while maintaining comparable mean integrated squared error rates. We investigate key asymptotic properties of our methodology and introduce a data-splitting technique to facilitate inference. The key attraction of our framework is its inference toolkit, which allows researchers to quantify uncertainty in causal discovery findings. We illustrate the performance of our methodology through simulation studies as well as multiple real data examples.

We introduce a semi-explicit time-stepping scheme of second order for linear poroelasticity satisfying a weak coupling condition. Here, semi-explicit means that the system, which needs to be solved in each step, decouples and hence improves the computational efficiency. The construction and the convergence proof are based on the connection to a differential equation with two time delays, namely one and two times the step size. Numerical experiments confirm the theoretical results and indicate the applicability to higher-order schemes.

1. Automated analysis of bioacoustic recordings using machine learning (ML) methods has the potential to greatly scale biodiversity monitoring efforts. The use of ML for high-stakes applications, such as conservation research, demands a data-centric approach with a focus on utilizing carefully annotated and curated evaluation and training data that is relevant and representative. Creating annotated datasets of sound recordings presents a number of challenges, such as managing large collections of recordings with associated metadata, developing flexible annotation tools that can accommodate the diverse range of vocalization profiles of different organisms, and addressing the scarcity of expert annotators. 2. We present Whombat a user-friendly, browser-based interface for managing audio recordings and annotation projects, with several visualization, exploration, and annotation tools. It enables users to quickly annotate, review, and share annotations, as well as visualize and evaluate a set of machine learning predictions on a dataset. The tool facilitates an iterative workflow where user annotations and machine learning predictions feedback to enhance model performance and annotation quality. 3. We demonstrate the flexibility of Whombat by showcasing two distinct use cases: an project aimed at enhancing automated UK bat call identification at the Bat Conservation Trust (BCT), and a collaborative effort among the USDA Forest Service and Oregon State University researchers exploring bioacoustic applications and extending automated avian classification models in the Pacific Northwest, USA. 4. Whombat is a flexible tool that can effectively address the challenges of annotation for bioacoustic research. It can be used for individual and collaborative work, hosted on a shared server or accessed remotely, or run on a personal computer without the need for coding skills.

In this paper, we develop reliable a posteriori error estimates for numerical approximations of scalar hyperbolic conservation laws in one space dimension. Our methods have no inherent small-data limitations and are a step towards error control of numerical schemes for systems. We are careful not to appeal to the Kruzhkov theory for scalar conservation laws. Instead, we derive novel quantitative stability estimates that extend the theory of shifts, and in particular, the framework for proving stability first developed by the second author and Vasseur. This is the first time this methodology has been used for quantitative estimates. We work entirely within the context of the theory of shifts and $a$-contraction, techniques which adapt well to systems. In fact, the stability framework by the second author and Vasseur has itself recently been pushed to systems [Chen-Krupa-Vasseur. Uniqueness and weak-BV stability for $2\times 2$ conservation laws. Arch. Ration. Mech. Anal., 246(1):299--332, 2022]. Our theoretical findings are complemented by a numerical implementation in MATLAB and numerical experiments.

This paper introduces an assumption-lean method that constructs valid and efficient lower predictive bounds (LPBs) for survival times with censored data. We build on recent work by Cand\`es et al. (2021), whose approach first subsets the data to discard any data points with early censoring times, and then uses a reweighting technique (namely, weighted conformal inference (Tibshirani et al., 2019)) to correct for the distribution shift introduced by this subsetting procedure. For our new method, instead of constraining to a fixed threshold for the censoring time when subsetting the data, we allow for a covariate-dependent and data-adaptive subsetting step, which is better able to capture the heterogeneity of the censoring mechanism. As a result, our method can lead to LPBs that are less conservative and give more accurate information. We show that in the Type I right-censoring setting, if either of the censoring mechanism or the conditional quantile of survival time is well estimated, our proposed procedure achieves nearly exact marginal coverage, where in the latter case we additionally have approximate conditional coverage. We evaluate the validity and efficiency of our proposed algorithm in numerical experiments, illustrating its advantage when compared with other competing methods. Finally, our method is applied to a real dataset to generate LPBs for users' active times on a mobile app.