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The phenomenom of emerging regular spectral features from random interactions is addressed in the context of the vibron model. A mean-field analysis links different regions of the parameter space with definite geometric shapes. The results…

In some fields of applications of stable distributions, especially in economics, it appears, that data have distributions similar to stable in a large region, but do not have such heavy tails. Our aim in this note is to propose several…

We study generalised linear regression and classification for a synthetically generated dataset encompassing different problems of interest, such as learning with random features, neural networks in the lazy training regime, and the hidden…

The study of multivariate extremes is dominated by multivariate regular variation, although it is well known that this approach does not provide adequate distinction between random vectors whose components are not always simultaneously…

In health cohort studies, repeated measures of markers are often used to describe the natural history of a disease. Joint models allow to study their evolution by taking into account the possible informative dropout usually due to clinical…

Since its introduction by J. Karamata, regular variation has evolved from a purely mathematical concept into a cornerstone of theoretical probability and data analysis. It is extensively studied and applied in different areas. Its…

We will outline novel approaches to derive model invariants for hidden Markov and related models. These approaches are based on a theoretical framework that arises from viewing random processes as elements of the vector space of string…

Nonlinear Mixed effects models are hidden variables models that are widely used in many fields such as pharmacometrics. In such models, the distribution characteristics of hidden variables can be specified by including several parameters…

Penalized regression methods are an attractive tool for high-dimensional data analysis, but their widespread adoption has been hampered by the difficulty of applying inferential tools. In particular, the question "How reliable is the…

The generalized perturbative approach is an all purpose variant of Stein's method used to obtain rates of normal approximation. Originally developed for functions of independent random variables this method is here extended to functions of…

Manifold learning is a popular and quickly-growing subfield of machine learning based on the assumption that one's observed data lie on a low-dimensional manifold embedded in a higher-dimensional space. This thesis presents a mathematical…

It is well known that the product of two independent regularly varying random variables with the same tail index is again regularly varying with this index. In this paper, we provide sharp sufficient conditions for the regular variation…

The well known domain shift issue causes model performance to degrade when deployed to a new target domain with different statistics to training. Domain adaptation techniques alleviate this, but need some instances from the target domain to…

Regular and rapid variation have been extensively studied in the literature and applied across various fields, particularly in extreme value theory. In this paper, we examine regular and rapid variation through the lens of generalized…

Real networks often form interacting parts of larger and more complex systems. Examples can be found in different domains, ranging from the Internet to structural and functional brain networks. Here, we show that these multiplex systems are…

The tail of the distribution of a sum of a random number of independent and identically distributed nonnegative random variables depends on the tails of the number of terms and of the terms themselves. This situation is of interest in the…

We derive properties of Latent Variable Models for networks, a broad class of models that includes the widely-used Latent Position Models. These include the average degree distribution, clustering coefficient, average path length and degree…

Online machine learning systems need to adapt to domain shifts. Meanwhile, acquiring label at every timestep is expensive. We propose a surprisingly simple algorithm that adaptively balances its regret and its number of label queries in…

There exist many forms of deep latent variable models, such as the variational autoencoder and adversarial autoencoder. Regardless of the specific class of model, there exists an implicit consensus that the latent distribution should be…

Here we introduce some new classes of discrete stable random variables, which are useful for understanding of a new general notion of stability of random variables called us as casual stability. There are given some examples of casual and…