# The Falling Factorial Basis and Its Statistical Applications

arXiv: https://arxiv.org/abs/1405.0558 · HTML: https://mlsanity.com/paper/1405.0558/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 3 May 2014.

## Built on by (most cited first, 8 of 8)

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- [A Cross Validation Framework for Signal Denoising with Applications to Trend Filtering, Dyadic CART and Beyond](https://mlsanity.com/paper/2201.02654/) (2022)
- [Distributed Learning with Discretely Observed Functional Data](https://mlsanity.com/paper/2410.02376/) (2024)
- [Data Fusion with Distributional Equivalence Test-then-pool](https://mlsanity.com/paper/2603.11867/) (2026)

## Abstract (arXiv)

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.