On constructing an approximation of some arbitrary function, and measuring the badness thereof.
THIS IS CHAOS RIGHT NOW. I need to break out the sampling/interpolation problem for regular data, for one thing.
Choosing the best approximation
In what sense? Most compact? Most easy to code?
If we are not interpolating, how much smoothing do we do?
We can use cross-validation, especially so-called “generalized” cross validation, to choose smoothing parameter this efficiently, in some sense.
Or you might have noisy data, in which case you now have a function approximation and inference problem, with error due to both approximation and sampling complexity. Compressive sensing can provide finite-sample guarantees for some of these settings.
To discuss: loss functions.
An interesting problem is how you align the curves that are your objects of study; That is a problem of warping.
Spline smoothing of observations
The classic.
Special superpowers: Easy to differentiate and integrate.
Special weakness: many free parameters, not so easy to do in high dimension.
See splines.
Polynomial bases
See polynomial bases.
Fourier bases
🏗️
Radial basis function approximation
I# Radial basis function approximation I actually care about this mostly for densities, so see mixture models, for what information I do have.
Rational approximation
Padé’s is the method I’ve heard of. Are there others? Easy to differentiate. OK to integrate if you cheat using a computational algebra package.
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