Divided Differences, Falling Factorials, and Discrete Splines: Another Look at Trend Filtering and Related Problems
Statistics Theory
2022-05-26 v3 Numerical Analysis
Numerical Analysis
Methodology
Statistics Theory
Abstract
This paper reviews a class of univariate piecewise polynomial functions known as discrete splines, which share properties analogous to the better-known class of spline functions, but where continuity in derivatives is replaced by (a suitable notion of) continuity in divided differences. As it happens, discrete splines bear connections to a wide array of developments in applied mathematics and statistics, from divided differences and Newton interpolation (dating back to over 300 years ago) to trend filtering (from the last 15 years). We survey these connections, and contribute some new perspectives and new results along the way.
Cite
@article{arxiv.2003.03886,
title = {Divided Differences, Falling Factorials, and Discrete Splines: Another Look at Trend Filtering and Related Problems},
author = {Ryan J. Tibshirani},
journal= {arXiv preprint arXiv:2003.03886},
year = {2022}
}
Comments
75 pages, 9 figures; 1 table