We study the eigenvalue distribution of random matrices pertinent to the analysis of deep neural ... more We study the eigenvalue distribution of random matrices pertinent to the analysis of deep neural networks. The matrices resemble the product of the sample covariance matrices, however, an important difference is that the analog of the population covariance matrix is now a function of random data matrices (synaptic weight matrices in the deep neural network terminology). The problem has been treated in recent work [J. Pennington, S. Schoenholz and S. Ganguli, The emergence of spectral universality in deep networks, Proc. Mach. Learn. Res. 84 (2018) 1924–1932, arXiv:1802.09979] by using the techniques of free probability theory. Since, however, free probability theory deals with population covariance matrices which are independent of the data matrices, its applicability in this case has to be justified. The justification has been given in [L. Pastur, On random matrices arising in deep neural networks: Gaussian case, Pure Appl. Funct. Anal. (2020), in press, arXiv:2001.06188] for Gauss...
We present a rigorous study of the perfect Bose-gas in the presence of a homogeneous ergodic rand... more We present a rigorous study of the perfect Bose-gas in the presence of a homogeneous ergodic random potential. It is demonstrated that the Lifshitz tail behaviour of the one-particle spectrum reduces the critical dimensionality of the (generalized) Bose–Einstein Condensation (BEC) to d = 1. To tackle the Off-Diagonal Long-Range Order (ODLRO) we introduce the space average one-body reduced density matrix. For a one-dimensional Poisson-type random potential we prove that randomness enhances the exponential decay of this matrix in domain free of the BEC. To cite this article: O. Lenoble et al., C. R. Physique 5 (2004).
Normalized eigenvalue counting measure of the sum of two Hermitian (or real symmetric) matrices A... more Normalized eigenvalue counting measure of the sum of two Hermitian (or real symmetric) matrices A n and B n rotated independently with respect to each other by the random unitary (or orthogonal) Haar distributed matrix U n (i.e. A n + U * n B n U n) is studied in the limit of large matrix order n. Convergence in probability to a limiting nonrandom measure is established. A functional equation for the Stieltjes transform of the limiting measure in terms of limiting eigenvalue measures of A n and B n is obtained and studied.
We discuss a method of the asymptotic computation of moments of the normalized eigenvalue countin... more We discuss a method of the asymptotic computation of moments of the normalized eigenvalue counting measure of random matrices of large order. The method is based on the resolvent identity and on some formulas relating expectations of certain matrix functions and the expectations including their derivatives or, equivalently, on some simple formulas of the perturbation theory. In the framework of
EQEC '05. European Quantum Electronics Conference, 2005., 2005
We report the observation of different localized structures coexisting for the same parameter val... more We report the observation of different localized structures coexisting for the same parameter values in an extended system. The experimental findings are carried out in a nonlinear optical interferometer, and are fully confirmed by numerical simulations. The existence of each kind of localized structure is put in relation to a corresponding delocalized pattern observed. Quantitative evaluation of the range of pump parameter allowing bistability between localized structures is given. The dependence of the phenomena on the other relevant parameters is discussed.
Eigenvalue problems play an important role in the dynamic analysis of engineering systems modeled... more Eigenvalue problems play an important role in the dynamic analysis of engineering systems modeled using the theory of linear structural mechanics. When uncertainties are considered, the eigenvalue problem becomes a random eigenvalue problem. In this paper the density of the eigenvalues of a discretized continuous system with uncertainty is discussed by considering the model where the system matrices are the Wishart random matrices. An analytical expression involving the Stieltjes transform is derived for the density of the ...
We consider the random dimer model in one space dimension with Bernoulli disorder. For sufficient... more We consider the random dimer model in one space dimension with Bernoulli disorder. For sufficiently small disorder, we show that the entanglement entropy exhibits at least a logarithmically enhanced area law if the Fermi energy coincides with a critical energy of the model where the localisation length diverges.
Journal of Physics A: Mathematical and Theoretical
We consider a random matrix model of interaction between a small n-level system, S, and its envir... more We consider a random matrix model of interaction between a small n-level system, S, and its environment, a N-level heat reservoir, R. The interaction between S and R is modeled by a tensor product of a fixed n×n matrix and a N ×N hermitian Gaussian random matrix. We show that under certain "macroscopicity" conditions on R, the reduced density matrix of the system ρ S = Tr R ρ (eq) S∪R , is given by ρ (c) S ∼ exp {−βH S }, where H S is the Hamiltonian of the isolated system. This holds for all strengths of the interaction and thus gives some justification for using ρ (c) S to describe some nanosystems, like biopolymers, in equilibrium with their environment [26]. Our results extend those obtained previously in [16, 17] for a special two-level system.
We present an experimental setup that provides the possibility to easily explore different kind o... more We present an experimental setup that provides the possibility to easily explore different kind of space-time chaos. This allows to extract and compare the relevant features of each of them, so as to further develop efficient adaptated control strategies. The system is an optical feedback loop closed through a Kerr-like non-linear medium (liquid crystal light valve). Beyond a critical value of the input light intensity (forcing), transverse instabilities develop in the optical beam. It results in the formation of out-of-equilibrium patterns, whose properties depend on various parameters that can be varied (translated or rotated feedback; focusing or defocusing non-linearity relative amount of diffraction and interferences, etc). Most of the time, the pattern at threshold is asymptotically stationary, after a diffusive transient. At higher forcing, patterns become non-stationary, and defects are usually created and annihilated continuously, giving rise to a space-time chaos regime. At even higher forcing, many lengthscales can be excited, and the regime becomes turbulent. We present preliminary results on such a space-time chaos characterisation using different analyzing tools (correlation functions, spectral analysis, Karhunen-Loeve decomposition, wavelet decomposition, mutual information...), as well as the different control strategies that might be implemented in each case.
We study the discrete Schrödinger operator H in Z d with the surface quasi periodic potential V (... more We study the discrete Schrödinger operator H in Z d with the surface quasi periodic potential V (x) = gδ(x 1) tan π(α • x 2 + ω), where x = (x 1 , x 2), x 1 ∈ Z d 1 , x 2 ∈ Z d 2 , α ∈ R d 2 , ω ∈ [0, 1). We first discuss a proof of the pure absolute continuity of the spectrum of H on the interval [−d, d] (the spectrum of the discrete Laplacian) in the case where the components of α are rationally independent. Then we show that in this case the generalized eigenfunctions have the form of the "volume" waves, i.e. of the sum of the incident plane wave and reflected from the hyper-plane Z d 1 waves, the form that is well known in the scattering theory for decaying potential. These eigenfunctions are orthogonal, complete and verify a natural analogue of the Lippmann-Schwinger equation. We find the wave operators and the scattering matrix in this case. We discuss also the case of rational α = p/q's, p, q ∈ N for d 1 = d 2 = 1, i.e. of a periodic surface potential. In this case besides the volume waves there are also the surface waves, whose amplitude decays exponentially as |x 1 | → ∞. For large q corresponding part of the absolutely continuos spectrum consists of q exponentially narrow bands, lying all except one outside the interval [−2, 2], and converging in a natural sense as q → ∞ to the dense point spectrum found before in [13] for the irrational Diophantine α's.
This paper adresses the behaviour of the mutual information of correlated MIMO Rayleigh channels ... more This paper adresses the behaviour of the mutual information of correlated MIMO Rayleigh channels when the numbers of transmit and receive antennas converge to +∞ at the same rate. Using a new and simple approach based on Poincaré-Nash inequality and on an integration by parts formula, it is rigorously established that the mutual information converges to a Gaussian random variable whose mean and variance are evaluated. These results confirm previous evaluations based on the powerful but non rigorous replica method. It is believed that the tools that are used in this paper are simple, robust, and of interest for the communications engineering community.
We study the eigenvalue distribution of random matrices pertinent to the analysis of deep neural ... more We study the eigenvalue distribution of random matrices pertinent to the analysis of deep neural networks. The matrices resemble the product of the sample covariance matrices, however, an important difference is that the analog of the population covariance matrix is now a function of random data matrices (synaptic weight matrices in the deep neural network terminology). The problem has been treated in recent work [J. Pennington, S. Schoenholz and S. Ganguli, The emergence of spectral universality in deep networks, Proc. Mach. Learn. Res. 84 (2018) 1924–1932, arXiv:1802.09979] by using the techniques of free probability theory. Since, however, free probability theory deals with population covariance matrices which are independent of the data matrices, its applicability in this case has to be justified. The justification has been given in [L. Pastur, On random matrices arising in deep neural networks: Gaussian case, Pure Appl. Funct. Anal. (2020), in press, arXiv:2001.06188] for Gauss...
We present a rigorous study of the perfect Bose-gas in the presence of a homogeneous ergodic rand... more We present a rigorous study of the perfect Bose-gas in the presence of a homogeneous ergodic random potential. It is demonstrated that the Lifshitz tail behaviour of the one-particle spectrum reduces the critical dimensionality of the (generalized) Bose–Einstein Condensation (BEC) to d = 1. To tackle the Off-Diagonal Long-Range Order (ODLRO) we introduce the space average one-body reduced density matrix. For a one-dimensional Poisson-type random potential we prove that randomness enhances the exponential decay of this matrix in domain free of the BEC. To cite this article: O. Lenoble et al., C. R. Physique 5 (2004).
Normalized eigenvalue counting measure of the sum of two Hermitian (or real symmetric) matrices A... more Normalized eigenvalue counting measure of the sum of two Hermitian (or real symmetric) matrices A n and B n rotated independently with respect to each other by the random unitary (or orthogonal) Haar distributed matrix U n (i.e. A n + U * n B n U n) is studied in the limit of large matrix order n. Convergence in probability to a limiting nonrandom measure is established. A functional equation for the Stieltjes transform of the limiting measure in terms of limiting eigenvalue measures of A n and B n is obtained and studied.
We discuss a method of the asymptotic computation of moments of the normalized eigenvalue countin... more We discuss a method of the asymptotic computation of moments of the normalized eigenvalue counting measure of random matrices of large order. The method is based on the resolvent identity and on some formulas relating expectations of certain matrix functions and the expectations including their derivatives or, equivalently, on some simple formulas of the perturbation theory. In the framework of
EQEC '05. European Quantum Electronics Conference, 2005., 2005
We report the observation of different localized structures coexisting for the same parameter val... more We report the observation of different localized structures coexisting for the same parameter values in an extended system. The experimental findings are carried out in a nonlinear optical interferometer, and are fully confirmed by numerical simulations. The existence of each kind of localized structure is put in relation to a corresponding delocalized pattern observed. Quantitative evaluation of the range of pump parameter allowing bistability between localized structures is given. The dependence of the phenomena on the other relevant parameters is discussed.
Eigenvalue problems play an important role in the dynamic analysis of engineering systems modeled... more Eigenvalue problems play an important role in the dynamic analysis of engineering systems modeled using the theory of linear structural mechanics. When uncertainties are considered, the eigenvalue problem becomes a random eigenvalue problem. In this paper the density of the eigenvalues of a discretized continuous system with uncertainty is discussed by considering the model where the system matrices are the Wishart random matrices. An analytical expression involving the Stieltjes transform is derived for the density of the ...
We consider the random dimer model in one space dimension with Bernoulli disorder. For sufficient... more We consider the random dimer model in one space dimension with Bernoulli disorder. For sufficiently small disorder, we show that the entanglement entropy exhibits at least a logarithmically enhanced area law if the Fermi energy coincides with a critical energy of the model where the localisation length diverges.
Journal of Physics A: Mathematical and Theoretical
We consider a random matrix model of interaction between a small n-level system, S, and its envir... more We consider a random matrix model of interaction between a small n-level system, S, and its environment, a N-level heat reservoir, R. The interaction between S and R is modeled by a tensor product of a fixed n×n matrix and a N ×N hermitian Gaussian random matrix. We show that under certain "macroscopicity" conditions on R, the reduced density matrix of the system ρ S = Tr R ρ (eq) S∪R , is given by ρ (c) S ∼ exp {−βH S }, where H S is the Hamiltonian of the isolated system. This holds for all strengths of the interaction and thus gives some justification for using ρ (c) S to describe some nanosystems, like biopolymers, in equilibrium with their environment [26]. Our results extend those obtained previously in [16, 17] for a special two-level system.
We present an experimental setup that provides the possibility to easily explore different kind o... more We present an experimental setup that provides the possibility to easily explore different kind of space-time chaos. This allows to extract and compare the relevant features of each of them, so as to further develop efficient adaptated control strategies. The system is an optical feedback loop closed through a Kerr-like non-linear medium (liquid crystal light valve). Beyond a critical value of the input light intensity (forcing), transverse instabilities develop in the optical beam. It results in the formation of out-of-equilibrium patterns, whose properties depend on various parameters that can be varied (translated or rotated feedback; focusing or defocusing non-linearity relative amount of diffraction and interferences, etc). Most of the time, the pattern at threshold is asymptotically stationary, after a diffusive transient. At higher forcing, patterns become non-stationary, and defects are usually created and annihilated continuously, giving rise to a space-time chaos regime. At even higher forcing, many lengthscales can be excited, and the regime becomes turbulent. We present preliminary results on such a space-time chaos characterisation using different analyzing tools (correlation functions, spectral analysis, Karhunen-Loeve decomposition, wavelet decomposition, mutual information...), as well as the different control strategies that might be implemented in each case.
We study the discrete Schrödinger operator H in Z d with the surface quasi periodic potential V (... more We study the discrete Schrödinger operator H in Z d with the surface quasi periodic potential V (x) = gδ(x 1) tan π(α • x 2 + ω), where x = (x 1 , x 2), x 1 ∈ Z d 1 , x 2 ∈ Z d 2 , α ∈ R d 2 , ω ∈ [0, 1). We first discuss a proof of the pure absolute continuity of the spectrum of H on the interval [−d, d] (the spectrum of the discrete Laplacian) in the case where the components of α are rationally independent. Then we show that in this case the generalized eigenfunctions have the form of the "volume" waves, i.e. of the sum of the incident plane wave and reflected from the hyper-plane Z d 1 waves, the form that is well known in the scattering theory for decaying potential. These eigenfunctions are orthogonal, complete and verify a natural analogue of the Lippmann-Schwinger equation. We find the wave operators and the scattering matrix in this case. We discuss also the case of rational α = p/q's, p, q ∈ N for d 1 = d 2 = 1, i.e. of a periodic surface potential. In this case besides the volume waves there are also the surface waves, whose amplitude decays exponentially as |x 1 | → ∞. For large q corresponding part of the absolutely continuos spectrum consists of q exponentially narrow bands, lying all except one outside the interval [−2, 2], and converging in a natural sense as q → ∞ to the dense point spectrum found before in [13] for the irrational Diophantine α's.
This paper adresses the behaviour of the mutual information of correlated MIMO Rayleigh channels ... more This paper adresses the behaviour of the mutual information of correlated MIMO Rayleigh channels when the numbers of transmit and receive antennas converge to +∞ at the same rate. Using a new and simple approach based on Poincaré-Nash inequality and on an integration by parts formula, it is rigorously established that the mutual information converges to a Gaussian random variable whose mean and variance are evaluated. These results confirm previous evaluations based on the powerful but non rigorous replica method. It is believed that the tools that are used in this paper are simple, robust, and of interest for the communications engineering community.
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Papers by Leonid Pastur