Function approximation and interpolation



A method of function approximation.

Special superpowers: Easy to differentiate and integrate.

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

References

Boyd, Nicholas, Trevor Hastie, Stephen Boyd, Benjamin Recht, and Michael Jordan. 2016. β€œSaturating Splines and Feature Selection.” arXiv:1609.06764 [Stat], September.
Cheney, Elliott Ward, and William Allan Light. 2009. A Course in Approximation Theory. American Mathematical Soc.
Dierckx, Paul. 1996. Curve and Surface Fitting Splines. Oxford: Clarendon Press.
Ekanadham, C., D. Tranchina, and E. P. Simoncelli. 2011. β€œRecovery of Sparse Translation-Invariant Signals With Continuous Basis Pursuit.” IEEE Transactions on Signal Processing 59 (10): 4735–44.
Fomel, Sergey. 2000. β€œInverse B-Spline Interpolation.” Citeseer.
Grohs, Philipp, Dmytro Perekrestenko, Dennis ElbrΓ€chter, and Helmut BΓΆlcskei. 2019. β€œDeep Neural Network Approximation Theory.” arXiv:1901.02220 [Cs, Math, Stat], January.
Hou, H.S., and H. Andrews. 1978. β€œCubic Splines for Image Interpolation and Digital Filtering.” IEEE Transactions on Acoustics, Speech and Signal Processing 26 (6): 508–17.
Poggio, T., and F. Girosi. 1990. β€œNetworks for Approximation and Learning.” Proceedings of the IEEE 78 (9): 1481–97.
Ramsay, J. O. 1988. β€œMonotone Regression Splines in Action.” Statistical Science 3 (4): 425–41.
Unser, M., A. Aldroubi, and M. Eden. 1993a. β€œB-Spline Signal Processing. I. Theory.” IEEE Transactions on Signal Processing 41 (2): 821–33.
β€”β€”β€”. 1993b. β€œB-Spline Signal Processing. II. Efficiency Design and Applications.” IEEE Transactions on Signal Processing 41 (2): 834–48.
Unser, Michael, Akram Aldroubi, and Murray Eden. 1991. β€œFast B-Spline Transforms for Continuous Image Representation and Interpolation.” IEEE Transactions on Pattern Analysis and Machine Intelligence 13 (3): 277–85.
Wang, Yu-Xiang, Alex Smola, and Ryan J. Tibshirani. 2014. β€œThe Falling Factorial Basis and Its Statistical Applications.” In Proceedings of the 31st International Conference on International Conference on Machine Learning - Volume 32, 730–38. ICML’14. Beijing, China: JMLR.org.
Weinert, H. L., and T. Kailath. 1974. β€œMinimum Energy Control Using Spline Functions.” In 1974 IEEE Conference on Decision and Control Including the 13th Symposium on Adaptive Processes, 169–72.
Wood, S. 1994. β€œMonotonic Smoothing Splines Fitted by Cross Validation.” SIAM Journal on Scientific Computing 15 (5): 1126–33.

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