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We study the spectrum of an asymmetric random matrix with block structured variances. The rows and columns of the random square matrix are divided into $D$ partitions with arbitrary size (linear in $N$). The parameters of the model are the…
High-dimensional sample correlation matrices are a crucial class of random matrices in multivariate statistical analysis. The central limit theorem (CLT) provides a theoretical foundation for statistical inference. In this paper, assuming…
Graph neural networks (GNNs) have attracted considerable attention from the research community. It is well established that GNNs are usually roughly divided into spatial and spectral methods. Despite that spectral GNNs play an important…
In recent years there has been a growing attention on distribution results in the sense of Weyl for the collective behavior of eigenvalues and singular values of matrix-sequences. Starting from the work of Szeg\"o regarding the case of…
Spectral clustering is one of the most popular unsupervised machine learning methods. Constructing similarity matrix is crucial to this type of method. In most existing works, the similarity matrix is computed once for all or is updated…
We consider a sequence of matrices that are associated to Markov dynamical systems and use determinant-free linear algebra techniques (as well as some algebra and complex analysis) to rigorously estimate the eigenvalues of every matrix…
In the present paper, we are concerned with the study of matrix-sequences arising from the discretization of PDEs and FDEs on domains $\Omega \subset \mathbb{R}^d$ with finite measure. When $\Omega$ is either a hypercube or a bounded…
This paper is devoted to the asymptotic behavior of all eigenvalues of Symmetric (in general non Hermitian) Toeplitz matrices with moderately smooth symbols which trace out a simple loop on the complex plane line as the dimension of the…
Large language models and deep neural networks achieve strong performance but suffer from reliability issues and high computational cost. This thesis proposes a unified framework based on spectral geometry and random matrix theory to…
We use random matrix theory to study the spectrum of random geometric graphs, a fundamental model of spatial networks. Considering ensembles of random geometric graphs we look at short range correlations in the level spacings of the…
We study the asymptotic eigenvalue distribution of Toeplitz matrices generated by a singular symbol. It has been conjectured by Widom that, for a generic symbol, the eigenvalues converge to the image of the symbol. In this paper we ask how…
Processing of symbolic sequences represented by mapping of symbolic data into numerical signals is commonly used in various applications. It is a particularly popular approach in genomic and proteomic sequence analysis. Numerous mappings of…
This article is concerned with the spectral behavior of $p$-dimensional linear processes in the moderately high-dimensional case when both dimensionality $p$ and sample size $n$ tend to infinity so that $p/n\to0$. It is shown that, under an…
Statistical inferences for sample correlation matrices are important in high dimensional data analysis. Motivated by this, this paper establishes a new central limit theorem (CLT) for a linear spectral statistic (LSS) of high dimensional…
In this paper, we study spectral properties of generalized weighted Hilbert matrices. In particular, we establish results on the spectral norm, determinant, as well as various relations between the eigenvalues and eigenvectors of such…
Dynamical networks are powerful tools for modeling a broad range of complex systems, including financial markets, brains, and ecosystems. They encode how the basic elements (nodes) of these systems interact altogether (via links) and evolve…
This work analyzes singular-value spectra of weight matrices in pretrained transformer models to understand how information is stored at both ends of the spectrum. Using Random Matrix Theory (RMT) as a zero information hypothesis, we…
Symbolic sequences generated by coupled map lattices (CMLs) can be used to model the chaotic-like structure of genomic sequences. In this study it is shown that diffusively coupled Chebyshev maps of order 4 (corresponding to a shift of 4…
We apply the method of spectral sequences to study classical problems in analysis. We illustrate the method by finding polynomial vector fields that commute with a given polynomial vector field and finding integrals of polynomial…
The singular value and spectral distribution of Toeplitz matrix sequences with Lebesgue integrable generating functions is well studied. Early results were provided in the classical Szeg{\H{o}} theorem and the Avram-Parter theorem, in which…