相关论文: Cavity and replica methods for the spectral densit…

The spectral density of various ensembles of sparse symmetric random matrices is analyzed using the cavity method. We consider two cases: matrices whose associated graphs are locally tree-like, and sparse covariance matrices. We derive a…

We show by direct calculation that the replica and cavity methods are exactly equivalent for the spectrum of Erdos-Renyi random graph. We introduce a variational formulation based on the cavity method and use it to find approximate…

We compute the spectral density for ensembles of of sparse symmetric random matrices using replica, managing to circumvent difficulties that have been encountered in earlier approaches along the lines first suggested in a seminal paper by…

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.…

The spectral density of random graphs with topological constraints is analysed using the replica method. We consider graph ensembles featuring generalised degree-degree correlations, as well as those with a community structure. In each case…

In physics, it is sometimes desirable to compute the so-called \emph{Density Of States} (DOS), also known as the \emph{spectral density}, of a real symmetric matrix $A$. The spectral density can be viewed as a probability density…

We derive the spectral properties of adjacency matrix of complex networks and of their Laplacian by the replica method combined with a dynamical population algorithm. By assuming the order parameter to be a product of Gaussian…

For random matrices with tree-like structure there exists a recursive relation for the local Green functions whose solution permits to find directly many important quantities in the limit of infinite matrix dimensions. The purpose of this…

In this work we study the spectral density of products of Wishart diluted random matrices of the form $X(1)\cdots X(M)(X(1)\cdots X(M))^T$ using the Edwards-Jones trick to map this problem into a system of interacting particles with random…

A methodology to analyze the properties of the first (largest) eigenvalue and its eigenvector is developed for large symmetric random sparse matrices utilizing the cavity method of statistical mechanics. Under a tree approximation, which is…

Sparse non-Hermitian random matrices arise in the study of disordered physical systems with asymmetric local interactions, and have applications ranging from neural networks to ecosystem dynamics. The spectral characteristics of these…

Theoretical analysis of biological and artificial neural networks e.g. modelling of synaptic or weight matrices necessitate consideration of the generic real-asymmetric matrix ensembles, those with varying order of matrix elements e.g. a…

We develop a theoretical approach to compute the conditioned spectral density of $N \times N$ non-invariant random matrices in the limit $N \rightarrow \infty$. This large deviation observable, defined as the eigenvalue distribution…

Penalized regression methods aim to retrieve reliable predictors among a large set of putative ones from a limited amount of measurements. In particular, penalized regression with singular penalty functions is important for sparse…

Applying the replica method of statistical mechanics, we evaluate the eigenvalue density of the large random matrix (sample covariance matrix) of the form $J = A^{\rm T} A$, where $A$ is an $M \times N$ real sparse random matrix. The…

We revisit the derivation of the density of states of sparse random matrices. We derive a recursion relation that allows one to compute the spectrum of the matrix of incidence for finite trees that determines completely the low…

In many machine learning and data related applications, it is required to have the knowledge of approximate ranks of large data matrices at hand. In this paper, we present two computationally inexpensive techniques to estimate the…

We study matchings on sparse random graphs by means of the cavity method. We first show how the method reproduces several known results about maximum and perfect matchings in regular and Erdos-Renyi random graphs. Our main new result is the…

Spectral clustering is a powerful method for finding structure in a dataset through the eigenvectors of a similarity matrix. It often outperforms traditional clustering algorithms such as $k$-means when the structure of the individual…

This paper studies sparse density estimation via $\ell_1$ penalization (SPADES). We focus on estimation in high-dimensional mixture models and nonparametric adaptive density estimation. We show, respectively, that SPADES can recover, with…