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The spectral densities of ensembles of non-Hermitian sparse random matrices are analysed using the cavity method. We present a set of equations from which the spectral density of a given ensemble can be efficiently and exactly calculated.…
We present a general method to detect and extract from a finite time sample statistically meaningful correlations between input and output variables of large dimensionality. Our central result is derived from the theory of free random…
Tensor models play an increasingly prominent role in many fields, notably in machine learning. In several applications, such as community detection, topic modeling and Gaussian mixture learning, one must estimate a low-rank signal from a…
We consider the moment space $\mathcal{M}_n$ corresponding to $p \times p$ real or complex matrix measures defined on the interval $[0,1]$. The asymptotic properties of the first $k$ components of a uniformly distributed vector $(S_{1,n},…
In this review paper we consider the polynuclear growth (PNG) model in one spatial dimension and its relation to random matrix ensembles. For curved and flat growth the scaling functions of the surface fluctuations coincide with limit…
The characteristic multi-dimensional integrals that represent physical quantities in random-matrix models, when calculated within the supersymmetry method, can be related to a class of integrals introduced in the context of two-dimensional…
We prove a local law in the bulk of the spectrum for random Gram matrices $XX^*$, a generalization of sample covariance matrices, where $X$ is a large matrix with independent, centered entries with arbitrary variances. The limiting…
The spectra of random feature matrices provide essential information on the conditioning of the linear system used in random feature regression problems and are thus connected to the consistency and generalization of random feature models.…
We numerically analyze the random matrix ensembles of real-symmetric matrices with column/row constraints for many system conditions e.g. disorder type, matrix-size and basis-connectivity. The results reveal a rich behavior hidden beneath…
The problem of finding large average submatrices of a real-valued matrix arises in the exploratory analysis of data from a variety of disciplines, ranging from genomics to social sciences. In this paper we provide a detailed asymptotic…
We investigate the spectra of adjacency matrices of multiplex networks under random matrix theory (RMT) framework. Through extensive numerical experiments, we demonstrate that upon multiplexing two random networks, the spectra of the…
Statistical properties of non--symmetric real random matrices of size $M$, obtained as truncations of random orthogonal $N\times N$ matrices are investigated. We derive an exact formula for the density of eigenvalues which consists of two…
We analyze statistical features of the ``optimization landscape'' in a random version of one of the simplest constrained optimization problems of the least-square type: finding the best approximation for the solution of an overcomplete…
We introduce a family of coefficients based on U-statistics that generalize the notion of correlation and explore their properties in the large dimensional multivariate case, showing that in the null case of uncorrelated variables, the…
A new method of estimating population linear spectral statistics from high-dimensional data is introduced. When the dimension $d$ grows with the sample size $n$ such that $\frac{d}{n} \to c>0$, the proposed method is the first with proven…
We consider an elastic manifold of internal dimension $d$ and length $L$ pinned in a $N$ dimensional random potential and confined by an additional parabolic potential of curvature $\mu$. We are interested in the mean spectral density…
We apply random matrix theory to complex networks. We show that nearest neighbor spacing distribution of the eigenvalues of the adjacency matrices of various model networks, namely scale-free, small-world and random networks follow…
A random phase property establishing a link between quasi-one-dimensional random Schroedinger operators and full random matrix theory is advocated. Briefly summarized it states that the random transfer matrices placed into a normal system…
These lectures provide an informal introduction into the notions and tools used to analyze statistical properties of eigenvalues of large random Hermitian matrices. After developing the general machinery of orthogonal polynomial method, we…
Spectral methods include a family of algorithms related to the eigenvectors of certain data-generated matrices. In this work, we are interested in studying the geometric landscape of the eigendecomposition problem in various spectral…