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 βgeneralizedβ 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.

## Polynomial spline smoothing of observations

The classic, and not just for functional data, but filed here because thatβs where the action is now.

Special superpowers: Easy to differentiate and integrate.

Special weakness: many free parameters, not easy to do in high dimension.

## Polynomial bases

See polynomial bases.

## Fourier bases

ποΈ

## 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

PadΓ©βs is the method Iβve heard of. Are there others? Easy to differentiate. OK to integrate if you cheat using a computational algebra package.

## References

*The Annals of Statistics*36 (1): 64β94.

*IEEE Transactions on Information Theory*39 (3): 930β45.

*2009 47th Annual Allerton Conference on Communication, Control, and Computing, Allerton 2009*, 36β43.

*Applied and Computational Harmonic Analysis*28 (2): 131β49.

*Signal Processing Conference, 2008 16th European*, 1β5. IEEE.

*SIAM Journal on Mathematics of Data Science*, February.

*Chebyshev & Fourier Spectral Methods*. Second Edition, Revised edition. Lecture Notes in Engineering. Berlin Heidelberg: Springer-Verlag.

*arXiv:1609.06764 [Stat]*, September.

*SIAM Journal on Numerical Analysis*22 (2): 386β400.

*A Course in Approximation Theory*. American Mathematical Soc.

*Mathematics of Control, Signals and Systems*2: 303β14.

*arXiv:1702.08489 [Cs, Stat]*, February.

*IEEE Transactions on Image Processing*7 (2): 141β54.

*Constructive Approximation*13 (1): 57β98.

*Curve and Surface Fitting Splines*. Oxford: Clarendon Press.

*Mathematics of Computation*52 (186): 471β94.

*IEEE Transactions on Signal Processing*59 (10): 4735β44.

*SIAM Journal on Numerical Analysis*17 (2): 238β46.

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

*IEEE Transactions on Signal Processing*47 (7): 1890β1902.

*1997 IEEE International Conference on Acoustics, Speech, and Signal Processing*, 3:2037β40. Munich, Germany: IEEE.

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

*IEEE Computational Science Engineering*2 (2): 50β61.

*arXiv:1901.02220 [Cs, Math, Stat]*, January.

*Journal of Computational and Applied Mathematics*39 (3): 287β94.

*Neural Networks*2 (5): 359β66.

*IEEE Transactions on Acoustics, Speech and Signal Processing*26 (6): 508β17.

*IEEE Transactions on Signal Processing*55 (7): 3760β72.

*Advances in Neural Information Processing Systems 27*, edited by Z. Ghahramani, M. Welling, C. Cortes, N. D. Lawrence, and K. Q. Weinberger, 27:1215β23. Curran Associates, Inc.

*arXiv:2010.01155 [Cs, Stat]*, October.

*arXiv:1612.04111 [Cs, Stat]*, December.

*Organised Sound*2 (03): 179β91.

*arXiv:1702.07028 [Cs]*.

*IEEE Transactions on Information Theory*42 (6): 2118β32.

*Conference Record of The Twenty-Seventh Asilomar Conference on Signals, Systems and Computers*, 40β44 vol.1.

*Acta Numerica*8 (January): 143β95.

*Proceedings of the IEEE*78 (9): 1481β97.

*Advances in Neural Information Processing Systems 29*, edited by D. D. Lee, M. Sugiyama, U. V. Luxburg, I. Guyon, and R. Garnett, 3360β68. Curran Associates, Inc.

*Statistical Science*3 (4): 425β41.

*IEEE Signal Processing Magazine*8 (4): 14β38.

*arXiv:1705.05502 [Cs, Stat]*, May.

*Proceedings of the IEEE*98 (6): 1045β57.

*arXiv:1707.04615 [Cs]*, July.

*arXiv:1602.04485 [Cs, Stat]*.

*PMLR*, 3387β93.

*Bulletin of the American Meteorological Society*79 (1): 61β78.

*IEEE Transactions on Signal Processing*41 (2): 821β33.

*IEEE Transactions on Signal Processing*41 (2): 834β48.

*IEEE Transactions on Pattern Analysis and Machine Intelligence*13 (3): 277β85.

*AeroSenseβ99*, 3723:28β31. International Society for Optics and Photonics.

*Proceedings of the 31st International Conference on International Conference on Machine Learning - Volume 32*, 730β38. ICMLβ14. Beijing, China: JMLR.org.

*IEEE Transactions on Information Theory*58 (8): 4969β92.

*1974 IEEE Conference on Decision and Control Including the 13th Symposium on Adaptive Processes*, 169β72.

*Proceedings of IEEE International Symposium on Information Theory*.

*SIAM Journal on Scientific Computing*15 (5): 1126β33.

*Neural Networks: The Official Journal of the International Neural Network Society*10 (1): 99β109.

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