相关论文: Machine learning the vanishing order of rational L…

Regularization techniques are widely employed in optimization-based approaches for solving ill-posed inverse problems in data analysis and scientific computing. These methods are based on augmenting the objective with a penalty function,…

For any periodic function $f:{\mathbb N} \to {\mathbb C}$ with period $q$, we study the Dirichlet series $L(s,f):=\sum_{n\geq 1} f(n)/n^s.$ It is well-known that this admits an analytic continuation to the entire complex plane except at…

1. The popularity of Machine learning (ML), Deep learning (DL), and Artificial intelligence (AI) has risen sharply in recent years. Despite this spike in popularity, the inner workings of ML and DL algorithms are often perceived as opaque,…

To comprehensively evaluate the mathematical reasoning capabilities of Large Language Models (LLMs), researchers have introduced abundant mathematical reasoning datasets. However, most existing datasets primarily focus on linear reasoning,…

We consider the problem of analyzing the heterogeneity of clustering distributions for multiple groups of observed data, each of which is indexed by a covariate value, and inferring global clusters arising from observations aggregated over…

Cellular automata (CA) dynamics are ordered in terms of two global parameters, computable {\sl a priori} from the description of rules. While one of them (activity) has been used before, the second one is new; it estimates the average…

Large language models (LLMs) have achieved state-of-the-art performance in a variety of tasks, but remain largely opaque in terms of their internal mechanisms. Understanding these mechanisms is crucial to improve their reasoning abilities.…

We introduce Deep Linear Discriminant Analysis (DeepLDA) which learns linearly separable latent representations in an end-to-end fashion. Classic LDA extracts features which preserve class separability and is used for dimensionality…

Extremes occur in stationary regularly varying time series as short periods with several large observations, known as extremal blocks. We study cluster statistics summarizing the behavior of functions acting on these extremal blocks.…

We propose a new method based on machine learning to \emph{play the devil's advocate} and investigate the impact of unknown systematic effects in a quantitative way. This method proceeds by reversing the measurement process and using the…

We consider the problem of exact reconstruction of univariate functions with jump discontinuities at unknown positions from their moments. These functions are assumed to satisfy an a priori unknown linear homogeneous differential equation…

The present paper provides exact expressions for the probability distributions of linear functionals of the two-parameter Poisson--Dirichlet process $\operatorname {PD}(\alpha,\theta)$. We obtain distributional results yielding exact forms…

Dislocation networks and their evolution are known to control the mechanical properties of metal samples. However, the lack of computationally efficient and statistically rigorous descriptors for such defect systems has hindered the…

Let $q\ge3$ be an integer, $\chi$ denote a Dirichlet character modulo $q$, for any real number $a\ge 0$, we define the generalized Dirichlet $L$-functions $$ L(s,\chi,a)=\sum_{n=1}^{\infty}\frac{\chi(n)}{(n+a)^s}, $$ where $s=\sigma+it$…

Recently popularized randomized methods for principal component analysis (PCA) efficiently and reliably produce nearly optimal accuracy --- even on parallel processors --- unlike the classical (deterministic) alternatives. We adapt one of…

Data normalization is crucial in machine learning, usually performed by subtracting the mean and dividing by standard deviation, or by rescaling to a fixed range. In copula theory, popular in finance, there is used normalization to…

Principal value integrals are associated to multi-valued rational differential forms with normal crossings support on a non-singular algebraic variety. We prove their vanishing on rational surfaces in the context of a conjecture of…

In the practical deployment of machine learning (ML) models, missing data represents a recurring challenge. Missing data is often addressed when training ML models. But missing data also needs to be addressed when deciding predictions and…

The Dirichlet form is a generalization of the Laplacian, heavily used in the study of many diffusion-like processes. In this paper we present a nonstandard representation theorem for the Dirichlet form, showing that the usual Dirichlet form…

In this article, we obtain the analytic continuation of the multiple shifted Lucas zeta function, multiple Lucas $L$-function associated to Dirichlet characters and additive characters. We then compute a complete list of exact singularities…