The Falling Factorial Basis and Its Statistical Applications

TL;DR

This paper introduces the falling factorial basis, a new spline-like basis that allows for fast computations and has useful properties, with applications in trend filtering and statistical testing.

Contribution

The paper proposes the falling factorial basis, enabling linear-time computations and extending spline-like functions to new statistical applications.

Findings

Allows rapid basis matrix multiplication and inversion

Applicable to trend filtering over arbitrary points

Useful for higher-order Kolmogorov-Smirnov tests

Abstract

We study a novel spline-like basis, which we name the "falling factorial basis", bearing many similarities to the classic truncated power basis. The advantage of the falling factorial basis is that it enables rapid, linear-time computations in basis matrix multiplication and basis matrix inversion. The falling factorial functions are not actually splines, but are close enough to splines that they provably retain some of the favorable properties of the latter functions. We examine their application in two problems: trend filtering over arbitrary input points, and a higher-order variant of the two-sample Kolmogorov-Smirnov test.