105 research outputs found
Spectra of large time-lagged correlation matrices from Random Matrix Theory
We analyze the spectral properties of large, time-lagged correlation matrices
using the tools of random matrix theory. We compare predictions of the
one-dimensional spectra, based on approaches already proposed in the
literature. Employing the methods of free random variables and diagrammatic
techniques, we solve a general random matrix problem, namely the spectrum of a
matrix , where is an Gaussian random
matrix and is \textit{any} , not necessarily symmetric
(Hermitian) matrix. As a particular application, we present the spectral
features of the large lagged correlation matrices as a function of the depth of
the time-lag. We also analyze the properties of left and right eigenvector
correlations for the time-lagged matrices. We positively verify our results by
the numerical simulations.Comment: 44 pages, 11 figures; v2 typos corrected, final versio
Probing non-orthogonality of eigenvectors in non-Hermitian matrix models: diagrammatic approach
Using large arguments, we propose a scheme for calculating the two-point
eigenvector correlation function for non-normal random matrices in the large
limit. The setting generalizes the quaternionic extension of free
probability to two-point functions. In the particular case of biunitarily
invariant random matrices, we obtain a simple, general expression for the
two-point eigenvector correlation function, which can be viewed as a further
generalization of the single ring theorem. This construction has some striking
similarities to the freeness of the second kind known for the Hermitian
ensembles in large . On the basis of several solved examples, we conjecture
two kinds of microscopic universality of the eigenvectors - one in the bulk,
and one at the rim. The form of the conjectured bulk universality agrees with
the scaling limit found by Chalker and Mehlig [JT Chalker, B Mehlig, PRL,
\textbf{81}, 3367 (1998)] in the case of the complex Ginibre ensemble.Comment: 20 pages + 4 pages of references, 12 figs; v2: typos corrected, refs
added; v3: more explanator
Narain transform for spectral deformations of random matrix models
We start from applying the general idea of spectral projection (suggested by Olshanski and Borodin and advocated by Tao) to the complex Wishart model. Combining the ideas of spectral projection with the insights from quantum mechanics, we derive in an effortless way all spectral properties of the complex Wishart model: first, the Marchenko-Pastur distribution interpreted as a Bohr-Sommerfeld quantization condition for the hydrogen atom; second, hard (Bessel), soft (Airy) and bulk (sine) microscopic kernels from properly rescaled radial Schrödinger equation for the hydrogen atom. Then, generalizing the ideas based on Schrödinger equation to the case when Hamiltonian is non-Hermitian, we propose an analogous construction for spectral projections of universal kernels for bi-orthogonal ensembles. In particular, we demonstrate that the Narain transform is a natural extension of the Hankel transform for the products of Wishart matrices, yielding an explicit form of the universal kernel at the hard edge. We also show how the change of variables of the rescaled kernel allows us to make the link to the universal kernel of the Muttalib-Borodin ensemble. The proposed construction offers a simple alternative to standard methods of derivation of microscopic kernels. Finally, we speculate, that a suitable extension of the Bochner theorem for Sturm-Liouville operators may provide an additional insight into the classification of microscopic universality classes in random matrix theory
Czy społeczeństwa śnią o sprawiedliwym pastwisku? Interpretacja „Umowy społecznej” Jana Jakuba Rousseau
In this paper I present an interpretation of Jean Jacques Rousseau's “Social Contract” as a normative theory of rational thinking of a political community. Using the understanding of Rousseau, suggested many times in the literature, as a theorist of group intentionality, I present the interpretation of the concept of “general will” as the intention of a group subject, which is a political community. Then I explain Rousseau’s normative remarks on democratic procedures through the prism of decision theory, in particular the relationship between the concept of general will and some concepts in game theory as well as Marquis de Condorcet’s “Jury Theorem”. I will show that the vision of fair democratic procedures proposed by Rousseau adequately reflects the conclusions drawn from the adduced theorems.W tej pracy prezentuję interpretację „Umowy społecznej” (1762) Jana Jakuba Rousseau jako normatywnej teorii racjonalnego myślenia wspólnoty politycznej. Używając wielokrotnie sugerowanego w literaturze (m.in. Tuomela 2013, Thompson 2017) rozumienia Rousseau jako teoretyka intencjonalności grupowej, przedstawiam interpretację pojęcia „woli powszechnej” jako intencji grupowego podmiotu, którym jest wspólnota polityczna. Następnie tłumaczę uwagi Rousseau dotyczące normatywnych procedur demokratycznych poprzez pryzmat normatywnej teorii decyzji, w szczególności związek pomiędzy pojęciem woli powszechnej a teorią gier wieloosobowych oraz tzw. „twierdzeniem o ławie przysięgłych” Marquisa de Condorceta. Pokażę, że proponowana przez Rousseau wizja sprawiedliwych procedur demokratycznych w adekwatny sposób oddaje wnioski płynące z przytaczanych twierdzeń
Unveiling the significance of eigenvectors in diffusing non-hermitian matrices by identifying the underlying Burgers dynamics
Following our recent letter, we study in detail an entry-wise diffusion of
non-hermitian complex matrices. We obtain an exact partial differential
equation (valid for any matrix size and arbitrary initial conditions) for
evolution of the averaged extended characteristic polynomial. The logarithm of
this polynomial has an interpretation of a potential which generates a Burgers
dynamics in quaternionic space. The dynamics of the ensemble in the large
is completely determined by the coevolution of the spectral density and a
certain eigenvector correlation function. This coevolution is best visible in
an electrostatic potential of a quaternionic argument built of two complex
variables, the first of which governs standard spectral properties while the
second unravels the hidden dynamics of eigenvector correlation function. We
obtain general large formulas for both spectral density and 1-point
eigenvector correlation function valid for any initial conditions. We exemplify
our studies by solving three examples, and we verify the analytic form of our
solutions with numerical simulations.Comment: 24 pages, 11 figure
Dynamical Isometry is Achieved in Residual Networks in a Universal Way for any Activation Function
We demonstrate that in residual neural networks (ResNets) dynamical isometry
is achievable irrespectively of the activation function used. We do that by
deriving, with the help of Free Probability and Random Matrix Theories, a
universal formula for the spectral density of the input-output Jacobian at
initialization, in the large network width and depth limit. The resulting
singular value spectrum depends on a single parameter, which we calculate for a
variety of popular activation functions, by analyzing the signal propagation in
the artificial neural network. We corroborate our results with numerical
simulations of both random matrices and ResNets applied to the CIFAR-10
classification problem. Moreover, we study the consequence of this universal
behavior for the initial and late phases of the learning processes. We conclude
by drawing attention to the simple fact, that initialization acts as a
confounding factor between the choice of activation function and the rate of
learning. We propose that in ResNets this can be resolved based on our results,
by ensuring the same level of dynamical isometry at initialization
Dysonian dynamics of the Ginibre ensemble
We study the time evolution of Ginibre matrices whose elements undergo
Brownian motion. The non-Hermitian character of the Ginibre ensemble binds the
dynamics of eigenvalues to the evolution of eigenvectors in a non-trivial way,
leading to a system of coupled nonlinear equations resembling those for
turbulent systems. We formulate a mathematical framework allowing simultaneous
description of the flow of eigenvalues and eigenvectors, and we unravel a
hidden dynamics as a function of new complex variable, which in the standard
description is treated as a regulator only. We solve the evolution equations
for large matrices and demonstrate that the non-analytic behavior of the
Green's functions is associated with a shock wave stemming from a Burgers-like
equation describing correlations of eigenvectors. We conjecture that the hidden
dynamics, that we observe for the Ginibre ensemble, is a general feature of
non-Hermitian random matrix models and is relevant to related physical
applications.Comment: 5 pages, 2 figure
From synaptic interactions to collective dynamics in random neuronal networks models: critical role of eigenvectors and transient behavior
The study of neuronal interactions is currently at the center of several
neuroscience big collaborative projects (including the Human Connectome, the
Blue Brain, the Brainome, etc.) which attempt to obtain a detailed map of the
entire brain matrix. Under certain constraints, mathematical theory can advance
predictions of the expected neural dynamics based solely on the statistical
properties of such synaptic interaction matrix. This work explores the
application of free random variables (FRV) to the study of large synaptic
interaction matrices. Besides recovering in a straightforward way known results
on eigenspectra of neural networks, we extend them to heavy-tailed
distributions of interactions. More importantly, we derive analytically the
behavior of eigenvector overlaps, which determine stability of the spectra. We
observe that upon imposing the neuronal excitation/inhibition balance, although
the eigenvalues remain unchanged, their stability dramatically decreases due to
strong non-orthogonality of associated eigenvectors. It leads us to the
conclusion that the understanding of the temporal evolution of asymmetric
neural networks requires considering the entangled dynamics of both
eigenvectors and eigenvalues, which might bear consequences for learning and
memory processes in these models. Considering the success of FRV analysis in a
wide variety of branches disciplines, we hope that the results presented here
foster additional application of these ideas in the area of brain sciences.Comment: 24 pages + 4 pages of refs, 8 figure
Non-orthogonal eigenvectors, fluctuation-dissipation relations and entropy production
Celebrated fluctuation-dissipation theorem (FDT) linking the response
function to time dependent correlations of observables measured in the
reference unperturbed state is one of the central results in equilibrium
statistical mechanics. In this letter we discuss an extension of the standard
FDT to the case when multidimensional matrix representing transition
probabilities is strictly non-normal. This feature dramatically modifies the
dynamics, by incorporating the effect of eigenvector non-orthogonality via the
associated overlap matrix of Chalker-Mehlig type. In particular, the rate of
entropy production per unit time is strongly enhanced by that matrix. We
suggest, that this mechanism has an impact on the studies of collective
phenomena in neural matrix models, leading, via transient behavior, to such
phenomena as synchronisation and emergence of the memory. We also expect, that
the described mechanism generating the entropy production is generic for wide
class of phenomena, where dynamics is driven by non-normal operators. For the
case of driving by a large Ginibre matrix the entropy production rate is
evaluated analytically, as well as for the Rajan-Abbott model for neural
networks.Comment: 3 figures, 8 pages. Important references added, calculation of
entropy production rates for Rajan-Abbott model of neural networks and for
Ginibre ensemble completed, title change
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