Related papers: Sensivity of the Hermite rank
Under long memory, the limit theorems for normalized sums of random variables typically involve a positive integer called "Hermite rank". There is a different limit for each Hermite rank. From a statistical point of view, however, we argue…
Using the theory of generalized Weierstrass transform, we show that the Hermite rank is identical to the power rank in the Gaussian case, and that an Hermite rank higher than one is unstable with respect to a level shift.
This article is dedicated to the following class of problems. Start with an $N\times N$ Hermitian matrix randomly picked from a matrix ensemble - the reference matrix. Applying a rank-$t$ perturbation to it, with $t$ taking the values $1\le…
Les perturbations de faible rang de matrices al\'eatoires ont \'et\'e au c{\oe}ur de nombreux travaux ces vingt derni\`eres ann\'ees. En particulier, les cas non-hermitiens, moins repr\'esent\'es dans la litt\'erature en r\`egle…
The notion of `stable rank' of a matrix is central to the analysis of randomized matrix algorithms, covariance estimation, deep neural networks, and recommender systems. We compare the properties of the stable rank and intrinsic dimension…
Latent variable models are well-known to suffer from rank deficiencies, causing problems with convergence and stability. Such problems are compounded in the "reduced-group split-ballot multitrait-multimethod model", which omits a set of…
We study the limit law of a vector made up of normalized sums of functions of long-range dependent stationary Gaussian series. Depending on the memory parameter of the Gaussian series and on the Hermite ranks of the functions, the resulting…
Motivated by classification, up to order isomorphism, of some dense subgroups of Euclidean space that are free of minimal rank, we obtain apparently new invariants for an equivalence relation (intermediate between Hermite and Smith) on…
A number of random matrix ensembles permitting exact determination of their eigenvalue and eigenvector statistics maintain this property under a rank $1$ perturbation. Considered in this review are the additive rank $1$ perturbation of the…
Given two random variables taking values in a bounded interval, we study whether one dominates the other in higher-order stochastic dominance depends on the reference interval in the model setting. We obtain two results. First, the…
A significant obstacle in the development of robust machine learning models is covariate shift, a form of distribution shift that occurs when the input distributions of the training and test sets differ while the conditional label…
We consider ensembles of Gaussian (Hermite) and Wishart (Laguerre) $N\times N$ hermitian matrices. We study the effect of finite rank perturbations of these ensembles by a source term. The rank $r$ of the perturbation corresponds to the…
In this paper we study how perturbing a matrix changes its non-negative rank. We prove that the non-negative rank is upper-semicontinuos and we describe some special families of perturbations. We show how our results relate to Statistics in…
The Non-Linear Sigma Model (NLSM) is an example of a field theory on a target space exhibiting intricate geometry. One remarkable characteristic of the NLSM is asymptotic freedom, which triggers interest in perturbative calculations. In the…
We compute the limiting eigenvalue statistics at the edge of the spectrum of large Hermitian random matrices perturbed by the addition of small rank deterministic matrices. To be more precise, we consider random Hermitian matrices with…
Despite large language models' (LLMs) recent advancements, their bias and hallucination issues persist, and their ability to offer consistent preferential rankings remains underexplored. This study investigates the capacity of LLMs to…
In this article we describe a new Hermite series based sequential estimator for the Spearman rank correlation coefficient and provide algorithms applicable in both the stationary and non-stationary settings. To treat the non-stationary…
Matrices of (approximate) low rank are pervasive in data science, appearing in recommender systems, movie preferences, topic models, medical records, and genomics. While there is a vast literature on how to exploit low rank structure in…
Non-Hermitian dynamics in quantum systems preserves the rank of the state density operator. Using this insight, we develop a geometric framework to describe its time evolution. In particular, we identify mutually orthogonal coherent and…
This study investigates pseudo-Hermitian quantum mechanics, where the Hamiltonian satisfies a modified Hermiticity condition. We extend the uncertainty relation for such systems, demonstrating its equivalence to the standard Hermitian case…