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We investigate the parameter estimation and prediction of two forms of the stochastic SIR model driven by small L\'{e}vy noise with time-dependent periodic transmission. We present consistency and rate of convergence results for the least-squares estimators. We include simulation studies using the method of projected gradient descent.

Cardiovascular diseases remain the leading global cause of mortality. Age is an important covariate whose effect is most easily investigated in a healthy cohort to properly distinguish the former from disease-related changes. Traditionally, most of such insights have been drawn from the analysis of electrocardiogram (ECG) feature changes in individuals as they age. However, these features, while informative, may potentially obscure underlying data relationships. In this paper we present the following contributions: (1) We employ a deep-learning model and a tree-based model to analyze ECG data from a robust dataset of healthy individuals across varying ages in both raw signals and ECG feature format. (2) We use explainable AI methods to identify the most discriminative ECG features across age groups.(3) Our analysis with tree-based classifiers reveals age-related declines in inferred breathing rates and identifies notably high SDANN values as indicative of elderly individuals, distinguishing them from younger adults. (4) Furthermore, the deep-learning model underscores the pivotal role of the P-wave in age predictions across all age groups, suggesting potential changes in the distribution of different P-wave types with age. These findings shed new light on age-related ECG changes, offering insights that transcend traditional feature-based approaches.

Soft targets combined with the cross-entropy loss have shown to improve generalization performance of deep neural networks on supervised classification tasks. The standard cross-entropy loss however assumes data to be categorically distributed, which may often not be the case in practice. In contrast, InfoNCE does not rely on such an explicit assumption but instead implicitly estimates the true conditional through negative sampling. Unfortunately, it cannot be combined with soft targets in its standard formulation, hindering its use in combination with sophisticated training strategies. In this paper, we address this limitation by proposing a principled loss function that is compatible with probabilistic targets. Our new soft target InfoNCE loss is conceptually simple, efficient to compute, and can be derived within the framework of noise contrastive estimation. Using a toy example, we demonstrate shortcomings of the categorical distribution assumption of cross-entropy, and discuss implications of sampling from soft distributions. We observe that soft target InfoNCE performs on par with strong soft target cross-entropy baselines and outperforms hard target NLL and InfoNCE losses on popular benchmarks, including ImageNet. Finally, we provide a simple implementation of our loss, geared towards supervised classification and fully compatible with deep classification model trained with cross-entropy.

Instruction tuning (IT) is widely used to teach pretrained large language models (LLMs) to follow arbitrary instructions, but is under-studied in multilingual settings. In this work, we conduct a systematic study of zero-shot cross-lingual transfer in IT, when an LLM is instruction-tuned on English-only data and then tested on user prompts in other languages. We advocate for the importance of evaluating various aspects of model responses in multilingual instruction following and investigate the influence of different model configuration choices. We find that cross-lingual transfer does happen successfully in IT even if all stages of model training are English-centric, but only if multiliguality is taken into account in hyperparameter tuning and with large enough IT data. English-trained LLMs are capable of generating correct-language, comprehensive and helpful responses in other languages, but suffer from low factuality and may occasionally have fluency errors.

Generalized cross-validation (GCV) is a widely-used method for estimating the squared out-of-sample prediction risk that employs a scalar degrees of freedom adjustment (in a multiplicative sense) to the squared training error. In this paper, we examine the consistency of GCV for estimating the prediction risk of arbitrary ensembles of penalized least-squares estimators. We show that GCV is inconsistent for any finite ensemble of size greater than one. Towards repairing this shortcoming, we identify a correction that involves an additional scalar correction (in an additive sense) based on degrees of freedom adjusted training errors from each ensemble component. The proposed estimator (termed CGCV) maintains the computational advantages of GCV and requires neither sample splitting, model refitting, or out-of-bag risk estimation. The estimator stems from a finer inspection of the ensemble risk decomposition and two intermediate risk estimators for the components in this decomposition. We provide a non-asymptotic analysis of the CGCV and the two intermediate risk estimators for ensembles of convex penalized estimators under Gaussian features and a linear response model. Furthermore, in the special case of ridge regression, we extend the analysis to general feature and response distributions using random matrix theory, which establishes model-free uniform consistency of CGCV.

In (Dzanic, J. Comp. Phys., 508:113010, 2024), a limiting approach for high-order discontinuous Galerkin schemes was introduced which allowed for imposing constraints on the solution continuously (i.e., everywhere within the element). While exact for linear constraint functionals, this approach only imposed a sufficient (but not the minimum necessary) amount of limiting for nonlinear constraint functionals. This short note shows how this limiting approach can be extended to allow exactness for general nonlinear quasiconcave constraint functionals through a nonlinear limiting procedure, reducing unnecessary numerical dissipation. Some examples are shown for nonlinear pressure and entropy constraints in the compressible gas dynamics equations, where both analytic and iterative approaches are used.

We propose a theoretically justified and practically applicable slice sampling based Markov chain Monte Carlo (MCMC) method for approximate sampling from probability measures on Riemannian manifolds. The latter naturally arise as posterior distributions in Bayesian inference of matrix-valued parameters, for example belonging to either the Stiefel or the Grassmann manifold. Our method, called geodesic slice sampling, is reversible with respect to the distribution of interest, and generalizes Hit-and-run slice sampling on $\mathbb{R}^{d}$ to Riemannian manifolds by using geodesics instead of straight lines. We demonstrate the robustness of our sampler's performance compared to other MCMC methods dealing with manifold valued distributions through extensive numerical experiments, on both synthetic and real data. In particular, we illustrate its remarkable ability to cope with anisotropic target densities, without using gradient information and preconditioning.

Relying on sheaf theory, we introduce the notions of projected barcodes and projected distances for multi-parameter persistence modules. Projected barcodes are defined as derived pushforward of persistence modules onto $\mathbb{R}$. Projected distances come in two flavors: the integral sheaf metrics (ISM) and the sliced convolution distances (SCD). We conduct a systematic study of the stability of projected barcodes and show that the fibered barcode is a particular instance of projected barcodes. We prove that the ISM and the SCD provide lower bounds for the convolution distance. Furthermore, we show that the $\gamma$-linear ISM and the $\gamma$-linear SCD which are projected distances tailored for $\gamma$-sheaves can be computed using TDA software dedicated to one-parameter persistence modules. Moreover, the time and memory complexity required to compute these two metrics are advantageous since our approach does not require computing nor storing an entire $n$-persistence module.

Learning interpretable representations of data generative latent factors is an important topic for the development of artificial intelligence. With the rise of the large multimodal model, it can align images with text to generate answers. In this work, we propose a framework to comprehensively explain each latent variable in the generative models using a large multimodal model. We further measure the uncertainty of our generated explanations, quantitatively evaluate the performance of explanation generation among multiple large multimodal models, and qualitatively visualize the variations of each latent variable to learn the disentanglement effects of different generative models on explanations. Finally, we discuss the explanatory capabilities and limitations of state-of-the-art large multimodal models.