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Attention is a smoothed cubic spline
CoRRWe highlight a perhaps important but hitherto unobserved insight: The attention module in a transformer is a smoothed cubic spline. Viewed in this manner, this mysterious but critical component of a transformer becomes a natural development of an old notion deeply entrenched in classical approximation theory.
Zehua Lai, Lek-Heng Lim, Yucong Liu
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MachinesIn industrial trajectory planning and real-time motion control, quintic B-splines are widely used for corner smoothing owing to their local support and high-order continuity. However, existing evaluation methods mainly rely on basis-function recursion or
Zhenyu Yin +6 more
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Irish Mathematical Society Bulletin, 1985 This brief review of curve smoothing by means of splines reports no new theoretical results. The following two papers are recommended to get more information: \textit{E. J. Wegman} and \textit{I. W. Wright}, J. Am. Stat. Assoc. 78, 351-365 (1983; Zbl 0534.62017), and the paper of \textit{R. L. Eubank}, see the foregoing review, Zbl 0569.62052.
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The weighted v-spline as a double knot B-spline [PDF]
, 1991 The local support basis representation of the ‘weighted v-spline’ is derived in terms of double knot cubic B-splines, so providing a convenient form for computing and analysing the ...
Gregory, JA, Zhou, J
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, 2008 This chapter promotes, details and exploits the fact that (univariate) splines, i.e., smooth piecewise polynomial functions, are weighted sums of B-splines. 1.
Spline Curves (p
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Research on Path Smoothing Method for Robot Scanning Measurement Based on Multiple Curves
ActuatorsAs the field of robotics advances swiftly, industrial automation has become prevalent in the realms of manufacturing and precision measurement. In robot measurement applications, the original path often originates from the discrete output of CAD models ...
Chen Chen +7 more
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Comparison of Smoothing and Truncated Spline Estimators in Estimating Blood [PDF]
, 2019 The functions, namely regression functions, which describe the relationship of more than one response variable observed at several values of the predictor variables in which there are correlations between responses can be estimated by using both ...
Fatmawati, .- +2 more
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An Improved Cp Criterion for Spline Smoothing
, 2012 [[abstract]]Spline smoothing is a popular technique for curve fitting, in which selection of the smoothing parameter is crucial. Many methods such as Mallows’ Cp, generalized maximum likelihood (GML), and the extended exponential (EE) criterion have been
Chen, Chun-Shu; Huang, Hsin-Cheng
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Spline Smoothing over Difficult Regions
We consider the problem of smoothing data on two-dimensional grids with holes or gaps. Such grids are often referred to as difficult regions. Since the data is not observed on these locations, the gap is not part of the domain.
Siem Jan Koopman, Soon Yip Wong
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