The Falling Factorial Basis and Its Statistical Applications

By Yu-Xiang Wang, Alex Smola, Robert Tibshirani. 2014

49 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Computer Vision and Pattern Recognition, Environmental Engineering

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Citations in the corpus

Additive models with trend filtering (2017, background): “:::;n: (12) Note that the columns of H(X j;k) follow the order of the sorted inputs S jX j, but the rows do not; however, for S jH(X j;k), both its rows and columns of follow the order of S jX j. FromWang et al. (2014), we know that (S jH(X j;k)) 1 = C(X j) 1 k! D (X j;k+1) ; for some matrix C(X j”
Fast and Flexible ADMM Algorithms for Trend Filtering (2014, background): “Tibshirani (2014) focuses on understanding trend ﬁltering estimates and their statistical properties, and relies on externally derived algorithms for comparisons of computational eﬃciency.”