# Function approximation and interpolation

June 9, 2016 — June 9, 2016

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.

## 1 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.

## 2 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.

## 3 Polynomial bases

See polynomial bases.

## 4 Fourier bases

🏗️

## 5 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.

## 6 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.

## 7 References

*IEEE Transactions on Information Theory*.

*The Annals of Statistics*.

*2009 47th Annual Allerton Conference on Communication, Control, and Computing, Allerton 2009*.

*Applied and Computational Harmonic Analysis*.

*IEEE Transactions on Signal Processing*.

*SIAM Journal on Mathematics of Data Science*.

*Chebyshev & Fourier Spectral Methods*. Lecture Notes in Engineering.

*arXiv:1609.06764 [Stat]*.

*SIAM Journal on Numerical Analysis*.

*A Course in Approximation Theory*.

*Mathematics of Control, Signals and Systems*.

*arXiv:1702.08489 [Cs, Stat]*.

*IEEE Transactions on Image Processing*.

*Constructive Approximation*.

*Acta Numerica*.

*Curve and Surface Fitting Splines*.

*Mathematics of Computation*.

*IEEE Transactions on Signal Processing*.

*IEEE Transactions on Information Theory*.

*SIAM Journal on Numerical Analysis*.

*1997 IEEE International Conference on Acoustics, Speech, and Signal Processing*.

*IEEE Workshop on Applications of Signal Processing to Audio and Acoustics*.

*1997 IEEE ASSP Workshop on Applications of Signal Processing to Audio and Acoustics, 1997*.

*IEEE Transactions on Signal Processing*.

*IEEE Computational Science Engineering*.

*IMA Journal of Numerical Analysis*.

*Journal of Computational and Applied Mathematics*.

*Neural Networks*.

*IEEE Transactions on Acoustics, Speech and Signal Processing*.

*IEEE Transactions on Signal Processing*.

*Advances in Neural Information Processing Systems 27*.

*arXiv:2010.01155 [Cs, Stat]*.

*arXiv:1612.04111 [Cs, Stat]*.

*Organised Sound*.

*IEEE Transactions on Information Theory*.

*arXiv:1702.07028 [Cs]*.

*Spline Methods*.

*Analysis and Applications*.

*Conference Record of The Twenty-Seventh Asilomar Conference on Signals, Systems and Computers*.

*Acta Numerica*.

*Proceedings of the IEEE*.

*Advances in Neural Information Processing Systems 29*.

*Statistical Science*.

*IEEE Signal Processing Magazine*.

*arXiv:1705.05502 [Cs, Stat]*.

*Proceedings of the IEEE*.

*Foundations and Trends® in Theoretical Computer Science*.

*arXiv:1707.04615 [Cs]*.

*arXiv:1602.04485 [Cs, Stat]*.

*PMLR*.

*Bulletin of the American Meteorological Society*.

*IEEE Transactions on Pattern Analysis and Machine Intelligence*.

*IEEE Transactions on Signal Processing*.

*IEEE Transactions on Signal Processing*.

*AeroSense’99*.

*Proceedings of the 31st International Conference on International Conference on Machine Learning - Volume 32*. ICML’14.

*IEEE Transactions on Information Theory*.

*1974 IEEE Conference on Decision and Control Including the 13th Symposium on Adaptive Processes*.

*Proceedings of IEEE International Symposium on Information Theory*.

*SIAM Journal on Scientific Computing*.

*Proceedings of the 34th International Conference on Neural Information Processing Systems*. NIPS’20.

*Neural Networks: The Official Journal of the International Neural Network Society*.