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We consider random matrix ensembles on the set of Hermitian matrices that are heavy tailed, in particular not all moments exist, and that are invariant under the conjugate action of the unitary group. The latter property entails that the eigenvectors are Haar distributed and, therefore, factorise from the eigenvalue statistics. We prove a classification for stable matrix ensembles of this kind of matrices represented in terms of matrices, their eigenvalues and their diagonal entries with the help of the classification of the multivariate stable distributions and the harmonic analysis on symmetric matrix spaces. Moreover, we identify sufficient and necessary conditions for their domains of attraction. To illustrate our findings we discuss for instance elliptical invariant random matrix ensembles and P\'olya ensembles, the latter playing a particular role in matrix convolutions. As a byproduct we generalise the derivative principle on the Hermitian matrices to general tempered distributions. This principle relates the joint probability density of the eigenvalues and the diagonal entries of the random matrix.

Transformers can efficiently learn in-context from example demonstrations. Most existing theoretical analyses studied the in-context learning (ICL) ability of transformers for linear function classes, where it is typically shown that the minimizer of the pretraining loss implements one gradient descent step on the least squares objective. However, this simplified linear setting arguably does not demonstrate the statistical efficiency of ICL, since the pretrained transformer does not outperform directly solving linear regression on the test prompt. In this paper, we study ICL of a nonlinear function class via transformer with nonlinear MLP layer: given a class of \textit{single-index} target functions $f_*(\boldsymbol{x}) = \sigma_*(\langle\boldsymbol{x},\boldsymbol{\beta}\rangle)$, where the index features $\boldsymbol{\beta}\in\mathbb{R}^d$ are drawn from a $r$-dimensional subspace, we show that a nonlinear transformer optimized by gradient descent (with a pretraining sample complexity that depends on the \textit{information exponent} of the link functions $\sigma_*$) learns $f_*$ in-context with a prompt length that only depends on the dimension of the distribution of target functions $r$; in contrast, any algorithm that directly learns $f_*$ on test prompt yields a statistical complexity that scales with the ambient dimension $d$. Our result highlights the adaptivity of the pretrained transformer to low-dimensional structures of the function class, which enables sample-efficient ICL that outperforms estimators that only have access to the in-context data.

The balanced incomplete block design (BIBD) problem is a difficult combinatorial problem with a large number of symmetries, which add complexity to its resolution. In this paper, we propose a dual (integer) problem representation that serves as an alternative to the classical binary formulation of the problem. We attack this problem incrementally: firstly, we propose basic algorithms (i.e. local search techniques and genetic algorithms) intended to work separately on the two different search spaces (i.e. binary and integer); secondly, we propose two hybrid schemes: an integrative approach (i.e. a memetic algorithm) and a collaborative model in which the previous methods work in parallel, occasionally exchanging information. Three distinct two-dimensional structures are proposed as communication topology among the algorithms involved in the collaborative model, as well as a number of migration and acceptance criteria for sending and receiving data. An empirical analysis comparing a large number of instances of our schemes (with algorithms possibly working on different search spaces and with/without symmetry breaking methods) shows that some of these algorithms can be considered the state of the art of the metaheuristic methods applied to finding BIBDs. Moreover, our cooperative proposal is a general scheme from which distinct algorithmic variants can be instantiated to handle symmetrical optimisation problems. For this reason, we have also analysed its key parameters, thereby providing general guidelines for the design of efficient/robust cooperative algorithms devised from our proposal.

We propose a tamed-adaptive Milstein scheme for stochastic differential equations in which the first-order derivatives of the coefficients are locally H\"older continuous of order $\alpha$. We show that the scheme converges in the $L_2$-norm with a rate of $(1+\alpha)/2$ over both finite intervals $[0, T]$ and the infinite interval $(0, +\infty)$, under certain growth conditions on the coefficients.

In pursuit of enhancing the comprehensive efficiency of production systems, our study focused on the joint optimization problem of scheduling and machine maintenance in scenarios where product rework occurs. The primary challenge lies in the interdependence between product \underline{q}uality, machine \underline{r}eliability, and \underline{p}roduction scheduling, compounded by the uncertainties from machine degradation and product quality, which is prevalent in sophisticated manufacturing systems. To address this issue, we investigated the dynamic relationship among these three aspects, named as QRP-co-effect. On this basis, we constructed an optimization model that integrates production scheduling, machine maintenance, and product rework decisions, encompassing the context of stochastic degradation and product quality uncertainties within a mixed-integer programming problem. To effectively solve this problem, we proposed a dual-module solving framework that integrates planning and evaluation for solution improvement via dynamic communication. By analyzing the structural properties of this joint optimization problem, we devised an efficient solving algorithm with an interactive mechanism that leverages \emph{in-situ} condition information regarding the production system's state and computational resources. The proposed methodology has been validated through comparative and ablation experiments. The experimental results demonstrated the significant enhancement of production system efficiency, along with a reduction in machine maintenance costs in scenarios involving rework.

In recent years, denoising diffusion models have become a crucial area of research due to their abundance in the rapidly expanding field of generative AI. While recent statistical advances have delivered explanations for the generation ability of idealised denoising diffusion models for high-dimensional target data, implementations introduce thresholding procedures for the generating process to overcome issues arising from the unbounded state space of such models. This mismatch between theoretical design and implementation of diffusion models has been addressed empirically by using a \emph{reflected} diffusion process as the driver of noise instead. In this paper, we study statistical guarantees of these denoising reflected diffusion models. In particular, we establish minimax optimal rates of convergence in total variation, up to a polylogarithmic factor, under Sobolev smoothness assumptions. Our main contributions include the statistical analysis of this novel class of denoising reflected diffusion models and a refined score approximation method in both time and space, leveraging spectral decomposition and rigorous neural network analysis.

In this paper we consider a class of conjugate discrete-time Riccati equations (CDARE), arising originally from the linear quadratic regulation problem for discrete-time antilinear systems. Recently, we have proved the existence of the maximal solution to the CDARE with a nonsingular control weighting matrix under the framework of the constructive method. Our contribution in the work is to finding another meaningful Hermitian solutions, which has received little attention in this topic. Moreover, we show that some extremal solutions cannot be attained at the same time, and almost (anti-)stabilizing solutions coincide with some extremal solutions. It is to be expected that our theoretical results presented in this paper will play an important role in the optimal control problems for discrete-time antilinear systems.

Accurate computation of robust estimates for extremal quantiles of empirical distributions is an essential task for a wide range of applicative fields, including economic policymaking and the financial industry. Such estimates are particularly critical in calculating risk measures, such as Growth-at-Risk (GaR). % and Value-at-Risk (VaR). This work proposes a conformal framework to estimate calibrated quantiles, and presents an extensive simulation study and a real-world analysis of GaR to examine its benefits with respect to the state of the art. Our findings show that CP methods consistently improve the calibration and robustness of quantile estimates at all levels. The calibration gains are appreciated especially at extremal quantiles, which are critical for risk assessment and where traditional methods tend to fall short. In addition, we introduce a novel property that guarantees coverage under the exchangeability assumption, providing a valuable tool for managing risks by quantifying and controlling the likelihood of future extreme observations.

Matrix perturbation bounds (such as Weyl and Davis-Kahan) are frequently used in many branches of mathematics. Most of the classical results in this area are optimal, in the worst case analysis. However, in modern applications, both the ground and the nose matrices frequently have extra structural properties. For instance, it is often assumed that the ground matrix is essentially low rank, and the nose matrix is random or pseudo-random. We aim to rebuild a part of perturbation theory, adapting to these modern assumptions. The key idea to to exploit the skewness between the leading eigenvectors of the ground matrix and the noise matrix. We will do this by combining the classical contour integration method with new combinatorial ideas, resulting in a new machinery, which has a wide range of applications. Our new bounds are optimal under mild assumptions, with direct applications to central problems in many different areas. Among others, we derive a sharp result for the perturbation of a low rank matrix with random perturbation, answering an open question in this area. Next, we derive new, optimal, results concerning covariance estimator of the spiked model, an important model in statistics, bridging two different directions of current research. Finally, we use our results on the perturbation of eigenspaces to derive new results concerning eigenvalues of deterministic and random matrices. In particular, we obtain new results concerning the outliers in the deformed Wigner model and the least singular value of random matrices with non-zero mean.

We present accurate and mathematically consistent formulations of a diffuse-interface model for two-phase flow problems involving rapid evaporation. The model addresses challenges including discontinuities in the density field by several orders of magnitude, leading to high velocity and pressure jumps across the liquid-vapor interface, along with dynamically changing interface topologies. To this end, we integrate an incompressible Navier-Stokes solver combined with a conservative level-set formulation and a regularized, i.e., diffuse, representation of discontinuities into a matrix-free adaptive finite element framework. The achievements are three-fold: First, we propose mathematically consistent definitions for the level-set transport velocity in the diffuse interface region by extrapolating the velocity from the liquid or gas phase. They exhibit superior prediction accuracy for the evaporated mass and the resulting interface dynamics compared to a local velocity evaluation, especially for strongly curved interfaces. Second, we show that accurate prediction of the evaporation-induced pressure jump requires a consistent, namely a reciprocal, density interpolation across the interface, which satisfies local mass conservation. Third, the combination of diffuse interface models for evaporation with standard Stokes-type constitutive relations for viscous flows leads to significant pressure artifacts in the diffuse interface region. To mitigate these, we propose to introduce a correction term for such constitutive model types. Through selected analytical and numerical examples, the aforementioned properties are validated. The presented model promises new insights in simulation-based prediction of melt-vapor interactions in thermal multiphase flows such as in laser-based powder bed fusion of metals.