13,800 research outputs found
Optimal Controller and Filter Realisations using Finite-precision, Floating- point Arithmetic.
The problem of reducing the fragility of digital controllers and filters
implemented using finite-precision, floating-point arithmetic is considered.
Floating-point arithmetic parameter uncertainty is multiplicative, unlike
parameter uncertainty resulting from fixed-point arithmetic. Based on first-
order eigenvalue sensitivity analysis, an upper bound on the eigenvalue
perturbations is derived. Consequently, open-loop and closed-loop eigenvalue
sensitivity measures are proposed. These measures are dependent upon the filter/
controller realization. Problems of obtaining the optimal realization with
respect to both the open-loop and the closed-loop eigenvalue sensitivity
measures are posed. The problem for the open-loop case is completely solved.
Solutions for the closed-loop case are obtained using non-linear programming.
The problems are illustrated with a numerical example
Convergence Theory of Learning Over-parameterized ResNet: A Full Characterization
ResNet structure has achieved great empirical success since its debut. Recent
work established the convergence of learning over-parameterized ResNet with a
scaling factor on the residual branch where is the network
depth. However, it is not clear how learning ResNet behaves for other values of
. In this paper, we fully characterize the convergence theory of gradient
descent for learning over-parameterized ResNet with different values of .
Specifically, with hiding logarithmic factor and constant coefficients, we show
that for gradient descent is guaranteed to converge to the
global minma, and especially when the convergence is irrelevant
of the network depth. Conversely, we show that for ,
the forward output grows at least with rate in expectation and then the
learning fails because of gradient explosion for large . This means the
bound is sharp for learning ResNet with arbitrary depth.
To the best of our knowledge, this is the first work that studies learning
ResNet with full range of .Comment: 31 page
Assessment of density functional methods with correct asymptotic behavior
Long-range corrected (LC) hybrid functionals and asymptotically corrected
(AC) model potentials are two distinct density functional methods with correct
asymptotic behavior. They are known to be accurate for properties that are
sensitive to the asymptote of the exchange-correlation potential, such as the
highest occupied molecular orbital energies and Rydberg excitation energies of
molecules. To provide a comprehensive comparison, we investigate the
performance of the two schemes and others on a very wide range of applications,
including the asymptote problems, self-interaction-error problems, energy-gap
problems, charge-transfer problems, and many others. The LC hybrid scheme is
shown to consistently outperform the AC model potential scheme. In addition, to
be consistent with the molecules collected in the IP131 database [Y.-S. Lin,
C.-W. Tsai, G.-D. Li, and J.-D. Chai, J. Chem. Phys., 2012, 136, 154109], we
expand the EA115 and FG115 databases to include, respectively, the vertical
electron affinities and fundamental gaps of the additional 16 molecules, and
develop a new database AE113 (113 atomization energies), consisting of accurate
reference values for the atomization energies of the 113 molecules in IP131.
These databases will be useful for assessing the accuracy of density functional
methods.Comment: accepted for publication in Phys. Chem. Chem. Phys., 46 pages, 4
figures, supplementary material include
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