63 research outputs found
Marcinkiewicz--Zygmund measures on manifolds
Let be a compact, connected, Riemannian manifold (without
boundary), be the geodesic distance on , be a
probability measure on , and be an orthonormal system
of continuous functions, for all ,
be an nondecreasing sequence of real numbers with
, as , , . We describe conditions to ensure an
equivalence between the norms of elements of with their suitably
discretized versions. We also give intrinsic criteria to determine if any
system of weights and nodes allows such inequalities. The results are stated in
a very general form, applicable for example, when the discretization of the
integrals is based on weighted averages of the elements of on geodesic
balls rather than point evaluations.Comment: 28 pages, submitted for publicatio
Bernstein-Markov type inequalities and discretization of norms
In this expository paper we will give a survey of some recent results concerning discretization of uniform and integral norms of polynomials and exponential sums which are based on various new Bernstein-Markov type inequalities. © 2021, Padova University Press. All rights reserved
Bypassing the quadrature exactness assumption of hyperinterpolation on the sphere
This paper focuses on the approximation of continuous functions on the unit
sphere by spherical polynomials of degree via hyperinterpolation.
Hyperinterpolation of degree is a discrete approximation of the
-orthogonal projection of degree with its Fourier coefficients
evaluated by a positive-weight quadrature rule that exactly integrates all
spherical polynomials of degree at most . This paper aims to bypass this
quadrature exactness assumption by replacing it with the Marcinkiewicz--Zygmund
property proposed in a previous paper. Consequently, hyperinterpolation can be
constructed by a positive-weight quadrature rule (not necessarily with
quadrature exactness). This scheme is referred to as unfettered
hyperinterpolation. This paper provides a reasonable error estimate for
unfettered hyperinterpolation. The error estimate generally consists of two
terms: a term representing the error estimate of the original
hyperinterpolation of full quadrature exactness and another introduced as
compensation for the loss of exactness degrees. A guide to controlling the
newly introduced term in practice is provided. In particular, if the quadrature
points form a quasi-Monte Carlo (QMC) design, then there is a refined error
estimate. Numerical experiments verify the error estimates and the practical
guide.Comment: 22 pages, 7 figure
Does generalization performance of regularization learning depend on ? A negative example
-regularization has been demonstrated to be an attractive technique in
machine learning and statistical modeling. It attempts to improve the
generalization (prediction) capability of a machine (model) through
appropriately shrinking its coefficients. The shape of a estimator
differs in varying choices of the regularization order . In particular,
leads to the LASSO estimate, while corresponds to the smooth
ridge regression. This makes the order a potential tuning parameter in
applications. To facilitate the use of -regularization, we intend to
seek for a modeling strategy where an elaborative selection on is
avoidable. In this spirit, we place our investigation within a general
framework of -regularized kernel learning under a sample dependent
hypothesis space (SDHS). For a designated class of kernel functions, we show
that all estimators for attain similar generalization
error bounds. These estimated bounds are almost optimal in the sense that up to
a logarithmic factor, the upper and lower bounds are asymptotically identical.
This finding tentatively reveals that, in some modeling contexts, the choice of
might not have a strong impact in terms of the generalization capability.
From this perspective, can be arbitrarily specified, or specified merely by
other no generalization criteria like smoothness, computational complexity,
sparsity, etc..Comment: 35 pages, 3 figure
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