3,085 research outputs found
Learning Kernel-Based Halfspaces with the Zero-One Loss
We describe and analyze a new algorithm for agnostically learning
kernel-based halfspaces with respect to the \emph{zero-one} loss function.
Unlike most previous formulations which rely on surrogate convex loss functions
(e.g. hinge-loss in SVM and log-loss in logistic regression), we provide finite
time/sample guarantees with respect to the more natural zero-one loss function.
The proposed algorithm can learn kernel-based halfspaces in worst-case time
\poly(\exp(L\log(L/\epsilon))), for \emph{any} distribution, where is a
Lipschitz constant (which can be thought of as the reciprocal of the margin),
and the learned classifier is worse than the optimal halfspace by at most
. We also prove a hardness result, showing that under a certain
cryptographic assumption, no algorithm can learn kernel-based halfspaces in
time polynomial in .Comment: This is a full version of the paper appearing in the 23rd
International Conference on Learning Theory (COLT 2010). Compared to the
previous arXiv version, this version contains some small corrections in the
proof of Lemma 3 and in appendix
On the constrained mock-Chebyshev least-squares
The algebraic polynomial interpolation on uniformly distributed nodes is
affected by the Runge phenomenon, also when the function to be interpolated is
analytic. Among all techniques that have been proposed to defeat this
phenomenon, there is the mock-Chebyshev interpolation which is an interpolation
made on a subset of the given nodes whose elements mimic as well as possible
the Chebyshev-Lobatto points. In this work we use the simultaneous
approximation theory to combine the previous technique with a polynomial
regression in order to increase the accuracy of the approximation of a given
analytic function. We give indications on how to select the degree of the
simultaneous regression in order to obtain polynomial approximant good in the
uniform norm and provide a sufficient condition to improve, in that norm, the
accuracy of the mock-Chebyshev interpolation with a simultaneous regression.
Numerical results are provided.Comment: 17 pages, 9 figure
A fast and well-conditioned spectral method for singular integral equations
We develop a spectral method for solving univariate singular integral
equations over unions of intervals by utilizing Chebyshev and ultraspherical
polynomials to reformulate the equations as almost-banded infinite-dimensional
systems. This is accomplished by utilizing low rank approximations for sparse
representations of the bivariate kernels. The resulting system can be solved in
operations using an adaptive QR factorization, where is
the bandwidth and is the optimal number of unknowns needed to resolve the
true solution. The complexity is reduced to operations by
pre-caching the QR factorization when the same operator is used for multiple
right-hand sides. Stability is proved by showing that the resulting linear
operator can be diagonally preconditioned to be a compact perturbation of the
identity. Applications considered include the Faraday cage, and acoustic
scattering for the Helmholtz and gravity Helmholtz equations, including
spectrally accurate numerical evaluation of the far- and near-field solution.
The Julia software package SingularIntegralEquations.jl implements our method
with a convenient, user-friendly interface
Numerical approximations for population growth model by Rational Chebyshev and Hermite Functions collocation approach: A comparison
This paper aims to compare rational Chebyshev (RC) and Hermite functions (HF)
collocation approach to solve the Volterra's model for population growth of a
species within a closed system. This model is a nonlinear integro-differential
equation where the integral term represents the effect of toxin. This approach
is based on orthogonal functions which will be defined. The collocation method
reduces the solution of this problem to the solution of a system of algebraic
equations. We also compare these methods with some other numerical results and
show that the present approach is applicable for solving nonlinear
integro-differential equations.Comment: 18 pages, 5 figures; Published online in the journal of "Mathematical
Methods in the Applied Sciences
An algorithm for the rapid numerical evaluation of Bessel functions of real orders and arguments
We describe a method for the rapid numerical evaluation of the Bessel
functions of the first and second kinds of nonnegative real orders and positive
arguments. Our algorithm makes use of the well-known observation that although
the Bessel functions themselves are expensive to represent via piecewise
polynomial expansions, the logarithms of certain solutions of Bessel's equation
are not. We exploit this observation by numerically precomputing the logarithms
of carefully chosen Bessel functions and representing them with piecewise
bivariate Chebyshev expansions. Our scheme is able to evaluate Bessel functions
of orders between and 1\sep,000\sep,000\sep,000 at essentially any
positive real argument. In that regime, it is competitive with existing methods
for the rapid evaluation of Bessel functions and has several advantages over
them. First, our approach is quite general and can be readily applied to many
other special functions which satisfy second order ordinary differential
equations. Second, by calculating the logarithms of the Bessel functions rather
than the Bessel functions themselves, we avoid many issues which arise from
numerical overflow and underflow. Third, in the oscillatory regime, our
algorithm calculates the values of a nonoscillatory phase function for Bessel's
differential equation and its derivative. These quantities are useful for
computing the zeros of Bessel functions, as well as for rapidly applying the
Fourier-Bessel transform. The results of extensive numerical experiments
demonstrating the efficacy of our algorithm are presented. A Fortran package
which includes our code for evaluating the Bessel functions as well as our code
for all of the numerical experiments described here is publically available
Fast, numerically stable computation of oscillatory integrals with stationary points
We present a numerically stable way to compute oscillatory integrals of the form . For each additional frequency, only a small, well-conditioned linear system with a Hessenberg matrix must be solved, and the amount of work needed decreases as the frequency increases. Moreover, we can modify the method for computing oscillatory integrals with stationary points. This is the first stable algorithm for oscillatory integrals with stationary points which does not lose accuracy as the frequency increases and does not require deformation into the complex plane
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