Divided Differences, Falling Factorials, and Discrete Splines: Another Look at Trend Filtering and Related Problems

TL;DR

This paper reviews discrete splines, a class of univariate piecewise polynomial functions, exploring their mathematical properties, historical connections, and relevance to trend filtering and related problems, offering new insights and results.

Contribution

It provides a comprehensive survey of discrete splines, highlighting their connections to classical and modern mathematical concepts, and introduces new perspectives and results.

Findings

Discrete splines connect to divided differences and Newton interpolation.

They relate to trend filtering and other applied statistical methods.

The paper offers new theoretical insights and perspectives.

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.