7,898 research outputs found
On Matching, and Even Rectifying, Dynamical Systems through Koopman Operator Eigenfunctions
Matching dynamical systems, through different forms of conjugacies and
equivalences, has long been a fundamental concept, and a powerful tool, in the
study and classification of nonlinear dynamic behavior (e.g. through normal
forms). In this paper we will argue that the use of the Koopman operator and
its spectrum is particularly well suited for this endeavor, both in theory, but
also especially in view of recent data-driven algorithm developments. We
believe, and document through illustrative examples, that this can nontrivially
extend the use and applicability of the Koopman spectral theoretical and
computational machinery beyond modeling and prediction, towards what can be
considered as a systematic discovery of "Cole-Hopf-type" transformations for
dynamics.Comment: 34 pages, 10 figure
Learning curves for Gaussian process regression: Approximations and bounds
We consider the problem of calculating learning curves (i.e., average
generalization performance) of Gaussian processes used for regression. On the
basis of a simple expression for the generalization error, in terms of the
eigenvalue decomposition of the covariance function, we derive a number of
approximation schemes. We identify where these become exact, and compare with
existing bounds on learning curves; the new approximations, which can be used
for any input space dimension, generally get substantially closer to the truth.
We also study possible improvements to our approximations. Finally, we use a
simple exactly solvable learning scenario to show that there are limits of
principle on the quality of approximations and bounds expressible solely in
terms of the eigenvalue spectrum of the covariance function.Comment: 25 pages, 10 figure
Spectral Theory of Sparse Non-Hermitian Random Matrices
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 matrices provide crucial information on system
stability and susceptibility, however, their study is greatly complicated by
the twin challenges of a lack of symmetry and a sparse interaction structure.
In this review we provide a concise and systematic introduction to the main
tools and results in this field. We show how the spectra of sparse
non-Hermitian matrices can be computed via an analogy with infinite dimensional
operators obeying certain recursion relations. With reference to three
illustrative examples --- adjacency matrices of regular oriented graphs,
adjacency matrices of oriented Erd\H{o}s-R\'{e}nyi graphs, and adjacency
matrices of weighted oriented Erd\H{o}s-R\'{e}nyi graphs --- we demonstrate the
use of these methods to obtain both analytic and numerical results for the
spectrum, the spectral distribution, the location of outlier eigenvalues, and
the statistical properties of eigenvectors.Comment: 60 pages, 10 figure
Eigenvalue Outliers of non-Hermitian Random Matrices with a Local Tree Structure
Spectra of sparse non-Hermitian random matrices determine the dynamics of
complex processes on graphs. Eigenvalue outliers in the spectrum are of
particular interest, since they determine the stationary state and the
stability of dynamical processes. We present a general and exact theory for the
eigenvalue outliers of random matrices with a local tree structure. For
adjacency and Laplacian matrices of oriented random graphs, we derive
analytical expressions for the eigenvalue outliers, the first moments of the
distribution of eigenvector elements associated with an outlier, the support of
the spectral density, and the spectral gap. We show that these spectral
observables obey universal expressions, which hold for a broad class of
oriented random matrices.Comment: 25 pages, 4 figure
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