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In the present work, we provide the general expression of the normalized centered moments of the Fr\'echet extreme-value distribution. In order to try to represent a set of data corresponding to rare events by a Fr\'echet distribution, it is important to be able to determine its characteristic parameter $\alpha$. Such a parameter can be deduced from the variance (proportional to the square of the Full Width at Half Maximum) of the studied distribution. However, the corresponding equation requires a numerical resolution. We propose two simple estimates of $\alpha$ from the knowledge of the variance, based on the Laurent series of the Gamma function. The most accurate expression involves the Ap\'ery constant.

Electronic health records (EHRs) store an extensive array of patient information, encompassing medical histories, diagnoses, treatments, and test outcomes. These records are crucial for enabling healthcare providers to make well-informed decisions regarding patient care. Summarizing clinical notes further assists healthcare professionals in pinpointing potential health risks and making better-informed decisions. This process contributes to reducing errors and enhancing patient outcomes by ensuring providers have access to the most pertinent and current patient data. Recent research has shown that incorporating prompts with large language models (LLMs) substantially boosts the efficacy of summarization tasks. However, we show that this approach also leads to increased output variance, resulting in notably divergent outputs even when prompts share similar meanings. To tackle this challenge, we introduce a model-agnostic Soft Prompt-Based Calibration (SPeC) pipeline that employs soft prompts to diminish variance while preserving the advantages of prompt-based summarization. Experimental findings on multiple clinical note tasks and LLMs indicate that our method not only bolsters performance but also effectively curbs variance for various LLMs, providing a more uniform and dependable solution for summarizing vital medical information.

This survey is devoted to recent developments in the statistical analysis of spherical data, with a view to applications in Cosmology. We will start from a brief discussion of Cosmological questions and motivations, arguing that most Cosmological observables are spherical random fields. Then, we will introduce some mathematical background on spherical random fields, including spectral representations and the construction of needlet and wavelet frames. We will then focus on some specific issues, including tools and algorithms for map reconstruction (\textit{i.e.}, separating the different physical components which contribute to the observed field), geometric tools for testing the assumptions of Gaussianity and isotropy, and multiple testing methods to detect contamination in the field due to point sources. Although these tools are introduced in the Cosmological context, they can be applied to other situations dealing with spherical data. Finally, we will discuss more recent and challenging issues such as the analysis of polarization data, which can be viewed as realizations of random fields taking values in spin fiber bundles.

We propose a method to fit arbitrarily accurate blendshape rig models by solving the inverse rig problem in realistic human face animation. The method considers blendshape models with different levels of added corrections and solves the regularized least-squares problem using coordinate descent, i.e., iteratively estimating blendshape weights. Besides making the optimization easier to solve, this approach ensures that mutually exclusive controllers will not be activated simultaneously and improves the goodness of fit after each iteration. We show experimentally that the proposed method yields solutions with mesh error comparable to or lower than the state-of-the-art approaches while significantly reducing the cardinality of the weight vector (over 20 percent), hence giving a high-fidelity reconstruction of the reference expression that is easier to manipulate in the post-production manually. Python scripts for the algorithm will be publicly available upon acceptance of the paper.

In addition to recent developments in computing speed and memory, methodological advances have contributed to significant gains in the performance of stochastic simulation. In this paper, we focus on variance reduction for matrix computations via matrix factorization. We provide insights into existing variance reduction methods for estimating the entries of large matrices. Popular methods do not exploit the reduction in variance that is possible when the matrix is factorized. We show how computing the square root factorization of the matrix can achieve in some important cases arbitrarily better stochastic performance. In addition, we propose a factorized estimator for the trace of a product of matrices and numerically demonstrate that the estimator can be up to 1,000 times more efficient on certain problems of estimating the log-likelihood of a Gaussian process. Additionally, we provide a new estimator of the log-determinant of a positive semi-definite matrix where the log-determinant is treated as a normalizing constant of a probability density.

Understanding variable dependence, particularly eliciting their statistical properties given a set of covariates, provides the mathematical foundation in practical operations management such as risk analysis and decision making given observed circumstances. This article presents an estimation method for modeling the conditional joint distribution of bivariate outcomes based on the distribution regression and factorization methods. This method is considered semiparametric in that it allows for flexible modeling of both the marginal and joint distributions conditional on covariates without imposing global parametric assumptions across the entire distribution. In contrast to existing parametric approaches, our method can accommodate discrete, continuous, or mixed variables, and provides a simple yet effective way to capture distributional dependence structures between bivariate outcomes and covariates. Various simulation results confirm that our method can perform similarly or better in finite samples compared to the alternative methods. In an application to the study of a motor third-part liability insurance portfolio, the proposed method effectively estimates risk measures such as the conditional Value-at-Risks and Expexted Sortfall. This result suggests that this semiparametric approach can serve as an alternative in insurance risk management.

We present EgoNeRF, a practical solution to reconstruct large-scale real-world environments for VR assets. Given a few seconds of casually captured 360 video, EgoNeRF can efficiently build neural radiance fields which enable high-quality rendering from novel viewpoints. Motivated by the recent acceleration of NeRF using feature grids, we adopt spherical coordinate instead of conventional Cartesian coordinate. Cartesian feature grid is inefficient to represent large-scale unbounded scenes because it has a spatially uniform resolution, regardless of distance from viewers. The spherical parameterization better aligns with the rays of egocentric images, and yet enables factorization for performance enhancement. However, the na\"ive spherical grid suffers from irregularities at two poles, and also cannot represent unbounded scenes. To avoid singularities near poles, we combine two balanced grids, which results in a quasi-uniform angular grid. We also partition the radial grid exponentially and place an environment map at infinity to represent unbounded scenes. Furthermore, with our resampling technique for grid-based methods, we can increase the number of valid samples to train NeRF volume. We extensively evaluate our method in our newly introduced synthetic and real-world egocentric 360 video datasets, and it consistently achieves state-of-the-art performance.

Tyler's and Maronna's M-estimators, as well as their regularized variants, are popular robust methods to estimate the scatter or covariance matrix of a multivariate distribution. In this work, we study the non-asymptotic behavior of these estimators, for data sampled from a distribution that satisfies one of the following properties: 1) independent sub-Gaussian entries, up to a linear transformation; 2) log-concave distributions; 3) distributions satisfying a convex concentration property. Our main contribution is the derivation of tight non-asymptotic concentration bounds of these M-estimators around a suitably scaled version of the data sample covariance matrix. Prior to our work, non-asymptotic bounds were derived only for Elliptical and Gaussian distributions. Our proof uses a variety of tools from non asymptotic random matrix theory and high dimensional geometry. Finally, we illustrate the utility of our results on two examples of practical interest: sparse covariance and sparse precision matrix estimation.

The memory hierarchy has a high impact on the performance and power consumption in the system. Moreover, current embedded systems, included in mobile devices, are specifically designed to run multimedia applications, which are memory intensive. This increases the pressure on the memory subsystem and affects the performance and energy consumption. In this regard, the thermal problems, performance degradation and high energy consumption, can cause irreversible damage to the devices. We address the optimization of the whole memory subsystem with three approaches integrated as a single methodology. Firstly, the thermal impact of register file is analyzed and optimized. Secondly, the cache memory is addressed by optimizing cache configuration according to running applications and improving both performance and power consumption. Finally, we simplify the design and evaluation process of general-purpose and customized dynamic memory manager, in the main memory. To this aim, we apply different evolutionary algorithms in combination with memory simulators and profiling tools. This way, we are able to evaluate the quality of each candidate solution and take advantage of the exploration of solutions given by the optimization algorithm.We also provide an experimental experience where our proposal is assessed using well-known benchmark applications.

Classic machine learning methods are built on the $i.i.d.$ assumption that training and testing data are independent and identically distributed. However, in real scenarios, the $i.i.d.$ assumption can hardly be satisfied, rendering the sharp drop of classic machine learning algorithms' performances under distributional shifts, which indicates the significance of investigating the Out-of-Distribution generalization problem. Out-of-Distribution (OOD) generalization problem addresses the challenging setting where the testing distribution is unknown and different from the training. This paper serves as the first effort to systematically and comprehensively discuss the OOD generalization problem, from the definition, methodology, evaluation to the implications and future directions. Firstly, we provide the formal definition of the OOD generalization problem. Secondly, existing methods are categorized into three parts based on their positions in the whole learning pipeline, namely unsupervised representation learning, supervised model learning and optimization, and typical methods for each category are discussed in detail. We then demonstrate the theoretical connections of different categories, and introduce the commonly used datasets and evaluation metrics. Finally, we summarize the whole literature and raise some future directions for OOD generalization problem. The summary of OOD generalization methods reviewed in this survey can be found at //out-of-distribution-generalization.com.