Spectral Analysis of Networks with Random Topologies

  title={Spectral Analysis of Networks with Random Topologies},
  author={Ulf Grenander and Jack W. Silverstein},
  journal={Siam Journal on Applied Mathematics},
A class of neural models is introduced in which the topology of the neural network has been generated by a controlled probability model. It is shown that the resulting linear operator has a spectral measure that converges in probability to a universal one when the size of the net tends to infinity: a law of large numbers for the spectra of such operators. The analytical treatment is accompanied by omputational experiments. 


A model for the generation of neural connections at birth led to the study of W, a random, symmetric, nonnegative definite linear operator defined on a finite, but very large, dimensional Euclidean

Asymptotics applied to a neural network

A mathematical model of neural processing is proposed which incorporates a theory for the storage of information that lends support to a distributive theory of memory using synaptic modification.

A New Method for Bounding Rates of Convergence of Empirical Spectral Distributions

The probabilistic properties of eigenvalues of random matrices whose dimension increases indefinitely has received considerable attention. One important aspect is the existence and identification of


In this paper, we give a brief review of the theory of spectral analysis of large dimensional random matrices. Most of the existing work in the literature has been stated for real matrices but the

Limiting Spectral Distributions of Large Dimensional Random Matrices

Models where the number of parameters increases with the sample size, are becoming increasingly important in statistics. This necessitates a close look at the statistical properties of eigenvalues of

Approximation of Haar distributed matrices and limiting distributions of eigenvalues of Jacobi ensembles

We develop a tool to approximate the entries of a large dimensional complex Jacobi ensemble with independent complex Gaussian random variables. Based on this and the author’s earlier work in this

On the empirical distribution of eigenvalues of a class of large dimensional random matrices

A stronger result on the limiting distribution of the eigenvalues of random Hermitian matrices of the form A + XTX*, originally studied in Marcenko and Pastur, is presented. Here, X(N - n), T(n - n),

Efficient Covariance Matrix Methods for Bayesian Gaussian Processes and Hopfield Neural Networks

It is shown using iterated function sequence methods that the new Hop eld learning rule acts as a palimpsest (or forgetful) learning rule, and does not suffer from catastrophic forgetting.



On the Distribution of the Roots of Certain Symmetric Matrices

The distribution law obtained before' for a very special set of matrices is valid for much more general sets of real symmetric matrices of very high dimensionality.

Asymptotic distribution of eigenvalues of random matrices

The impetus for this paper comes mainly from work done in recent years by a number of physicists on a statistical theory of spectra. The book by M. L. Mehta [10] and the collection of reprints edited

Characteristic Vectors of Bordered Matrices with Infinite Dimensions I

The statistical properties of the characteristic values of a matrix the elements of which show a normal (Gaussian) distribution are well known (cf. [6] Chapter XI) and have been derived, rather

A Brownian‐Motion Model for the Eigenvalues of a Random Matrix

A new type of Coulomb gas is defined, consisting of n point charges executing Brownian motions under the influence of their mutual electrostatic repulsions. It is proved that this gas gives an exact

Toeplitz Forms And Their Applications

Part I: Toeplitz Forms: Preliminaries Orthogonal polynomials. Algebraic properties Orthogonal polynomials. Limit properties The trigonometric moment problem Eigenvalues of Toeplitz forms

Theory of Functions

THE criticism on the passage quoted from p. 3 of the book by Profs. Harkness and Morley (NATURE, February 23, p. 347) turns on the fact that, in dealing with number divorced from measurement, the

Probabilities on Algebraic Structures, Almqvist and Wiksell

  • Probabilities on Algebraic Structures, Almqvist and Wiksell
  • 1963

Limit theorems ]:or the characteristic roots o]: a sample covariance matrix

  • Soviet Math. Dokl
  • 1971