TY - JOUR
T1 - High order Parzen windows and randomized sampling
AU - Zhou, Xiang-Jun
AU - Zhou, Ding-Xuan
PY - 2009/10
Y1 - 2009/10
N2 - In this paper high order Parzen windows stated by means of basic window functions are studied for understanding some algorithms in learning theory and randomized sampling in multivariate approximation. Learning rates are derived for the least-square regression and density estimation on bounded domains under some decay conditions on the marginal distributions near the boundary. These rates can be almost optimal when the marginal distributions decay fast and the order of the Parzen windows is large enough. For randomized sampling in shift-invariant spaces, we consider the situation when the sampling points are neither i.i.d. nor regular, but are noised from regular grids by probability density functions. The approximation orders are estimated by means of the regularity of the approximated function and the density function and the order of the Parzen windows. © Springer Science+Business Media, LLC 2008.
AB - In this paper high order Parzen windows stated by means of basic window functions are studied for understanding some algorithms in learning theory and randomized sampling in multivariate approximation. Learning rates are derived for the least-square regression and density estimation on bounded domains under some decay conditions on the marginal distributions near the boundary. These rates can be almost optimal when the marginal distributions decay fast and the order of the Parzen windows is large enough. For randomized sampling in shift-invariant spaces, we consider the situation when the sampling points are neither i.i.d. nor regular, but are noised from regular grids by probability density functions. The approximation orders are estimated by means of the regularity of the approximated function and the density function and the order of the Parzen windows. © Springer Science+Business Media, LLC 2008.
KW - Approximation
KW - Basic window function
KW - High order kernels
KW - Parzen windows
KW - Randomized sampling
KW - Regression
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UR - https://www.scopus.com/record/pubmetrics.uri?eid=2-s2.0-70350722408&origin=recordpage
U2 - 10.1007/s10444-008-9073-8
DO - 10.1007/s10444-008-9073-8
M3 - 21_Publication in refereed journal
VL - 31
SP - 349
EP - 368
JO - Advances in Computational Mathematics
JF - Advances in Computational Mathematics
SN - 1019-7168
IS - 4
ER -