Related papers: Superintegrability of matrix Student's distributio…

Real-world data is laden with outlying values. The challenge for machine learning is that the learner typically has no prior knowledge of whether the feedback it receives (losses, gradients, etc.) will be heavy-tailed or not. In this work,…

Risk assessment for rare events is essential for understanding systemic stability in complex systems. As rare events are typically highly correlated, it is important to study heavy-tailed multivariate distributions of the relevant…

Understanding random open quantum systems is critical for characterizing the performance of large-scale quantum devices and exploring macroscopic quantum phenomena. Various features in these systems, including spectral distributions, gap…

We provide some asymptotic theory for the largest eigenvalues of a sample covariance matrix of a p-dimensional time series where the dimension p = p_n converges to infinity when the sample size n increases. We give a short overview of the…

Let $X$ be a $n\times p$ matrix with coherence $\mu(X)=\max_{j\neq j'} |X_j^tX_{j'}|$. We present a simplified and improved study of the quasi-isometry property for most submatrices of $X$ obtained by uniform column sampling. Our results…

Random matrix theory allows for the deduction of stability criteria for complex systems using only a summary knowledge of the statistics of the interactions between components. As such, results like the well-known elliptical law are…

The dominant approaches to text representation in natural language rely on learning embeddings on massive corpora which have convenient properties such as compositionality and distance preservation. In this paper, we develop a novel method…

Some eigenvalue matrix models possess an interesting property: one can manifestly define the basis where all averages can be explicitly calculated. For example, in the Gaussian Hermitian and rectangular complex models, averages of the Schur…

Modern risk modelling approaches deal with vectors of multiple components. The components could be, for example, returns of financial instruments or losses within an insurance portfolio concerning different lines of business. One of the…

In this paper, we study tail inequalities of the largest eigenvalue of a matrix infinitely divisible (i.d.) series, which is a finite sum of fixed matrices weighted by i.d. random variables. We obtain several types of tail inequalities,…

Generalized linear mixed models are powerful tools for analyzing clustered data, where the unknown parameters are classically (and most commonly) estimated by the maximum likelihood and restricted maximum likelihood procedures. However,…

Extreme values and the tail behavior of probability distributions are essential for quantifying and mitigating risk in complex systems of all kinds. In multivariate settings, accounting for correlations is crucial. Although extreme value…

We re-visit tail the index regressions framework. For linear specifications, we find that the usual full rank condition can fail because conditioning on extreme outcomes causes regressors to degenerate to constants. Taking this into…

Real-world data are long-tailed, the lack of tail samples leads to a significant limitation in the generalization ability of the model. Although numerous approaches of class re-balancing perform well for moderate class imbalance problems,…

Extreme U-statistics arise when the kernel of a U-statistic has a high degree but depends only on its arguments through a small number of top order statistics. As the kernel degree of the U-statistic grows to infinity with the sample size,…

Convolutional neural networks have achieved great improvement on face recognition in recent years because of its extraordinary ability in learning discriminative features of people with different identities. To train such a well-designed…

The role of memorization in machine learning (ML) has garnered significant attention, particularly as modern models are empirically observed to memorize fragments of training data. Previous theoretical analyses, such as Feldman's seminal…

We show that a simple mechanistic model of spatial dispersal for settling organisms, subject to parameter variability, can generate heavy-tailed radial probability density functions. The movement of organisms in the model consists of a…

Random matrix models based on an integral over supermatrices are proposed as a natural extension of bosonic matrix models. The subtle nature of superspace integration allows these models to have very different properties from the analogous…

Modern image classifiers perform well on populated classes, while degrading considerably on tail classes with only a few instances. Humans, by contrast, effortlessly handle the long-tailed recognition challenge, since they can learn the…