39,107 research outputs found
Smooth eigenvalue correction
Second-order statistics play an important role in data modeling. Nowadays, there is a tendency toward measuring more signals with higher resolution (e.g., high-resolution video), causing a rapid increase of dimensionality of the measured samples, while the number of samples remains more or less the same. As a result the eigenvalue estimates are significantly biased as described by the MarÄŤenko Pastur equation for the limit of both the number of samples and their dimensionality going to infinity. By introducing a smoothness factor, we show that the MarÄŤenko Pastur equation can be used in practical situations where both the number of samples and their dimensionality remain finite.\ud
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Based on this result we derive methods, one already known and one new to our knowledge, to estimate the sample eigenvalues when the population eigenvalues are known. However, usually the sample eigenvalues are known and the population eigenvalues are required. We therefore applied one of the these methods in a feedback loop, resulting in an eigenvalue bias correction method.\ud
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We compare this eigenvalue correction method with the state-of-the-art methods and show that our method outperforms other methods particularly in real-life situations often encountered in biometrics: underdetermined configurations, high-dimensional configurations, and configurations where the eigenvalues are exponentially distributed
Discretization error cancellation in the plane-wave approximation of periodic Hamiltonians with Coulomb singularities
In solid-state physics, energies of molecular systems are usually computed
with a plane-wave discretization of Kohn-Sham equations. A priori estimates of
plane-wave convergence for periodic Kohn-Sham calculations with
pseudopotentials have been proved , however in most computations in practice,
plane-wave cut-offs are not tight enough to target the desired accuracy. It is
often advocated that the real quantity of interest is not the value of the
energy but of energy differences for different configurations. The computed
energy difference is believed to be much more accurate because of
`discretization error cancellation', since the sources of numerical errors are
essentially the same for different configurations. For periodic linear
Hamiltonians with Coulomb potentials, error cancellation can be explained by
the universality of the Kato cusp condition. Using weighted Sobolev spaces,
Taylor-type expansions of the eigenfunctions are available yielding a precise
characterization of this singularity. This then gives an explicit formula of
the first order term of the decay of the Fourier coefficients of the
eigenfunctions. It enables one to prove that errors on eigenvalue differences
are reduced but converge at the same rate as the error on the eigenvalue.Comment: 14 pages, 3 figures, improved result on main theorem, corrected typo
Conformally maximal metrics for Laplace eigenvalues on surfaces
The paper is concerned with the maximization of Laplace eigenvalues on
surfaces of given volume with a Riemannian metric in a fixed conformal class. A
significant progress on this problem has been recently achieved by
Nadirashvili-Sire and Petrides using related, though different methods. In
particular, it was shown that for a given , the maximum of the -th
Laplace eigenvalue in a conformal class on a surface is either attained on a
metric which is smooth except possibly at a finite number of conical
singularities, or it is attained in the limit while a "bubble tree" is formed
on a surface. Geometrically, the bubble tree appearing in this setting can be
viewed as a union of touching identical round spheres. We present another proof
of this statement, developing the approach proposed by the second author and Y.
Sire. As a side result, we provide explicit upper bounds on the topological
spectrum of surfaces.Comment: 52 pages, 3 figures, added a section on explicit constant in
Korevaar's inequality, minor correction
Eigenvalue distributions from a star product approach
We use the well-known isomorphism between operator algebras and function
spaces equipped with a star product to study the asymptotic properties of
certain matrix sequences in which the matrix dimension tends to infinity.
Our approach is based on the coherent states which allow for a
systematic 1/D expansion of the star product. This produces a trace formula for
functions of the matrix sequence elements in the large- limit which includes
higher order (finite-) corrections. From this a variety of analytic results
pertaining to the asymptotic properties of the density of states, eigenstates
and expectation values associated with the matrix sequence follows. It is shown
how new and existing results in the settings of collective spin systems and
orthogonal polynomial sequences can be readily obtained as special cases. In
particular, this approach allows for the calculation of higher order
corrections to the zero distributions of a large class of orthogonal
polynomials.Comment: 25 pages, 8 figure
Trace formulas for stochastic evolution operators: Smooth conjugation method
The trace formula for the evolution operator associated with nonlinear
stochastic flows with weak additive noise is cast in the path integral
formalism. We integrate over the neighborhood of a given saddlepoint exactly by
means of a smooth conjugacy, a locally analytic nonlinear change of field
variables. The perturbative corrections are transfered to the corresponding
Jacobian, which we expand in terms of the conjugating function, rather than the
action used in defining the path integral. The new perturbative expansion which
follows by a recursive evaluation of derivatives appears more compact than the
standard Feynman diagram perturbation theory. The result is a stochastic analog
of the Gutzwiller trace formula with the ``hbar'' corrections computed an order
higher than what has so far been attainable in stochastic and
quantum-mechanical applications.Comment: 16 pages, 1 figure, New techniques and results for a problem we
considered in chao-dyn/980703
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