Interpolation in Polynomial Spaces of p-Degree

Abstract: We recently introduced the Fast Newton Transform (FNT), an algorithm for performing multivariate Newton interpolation in downward closed polynomial spaces of spatial dimension $m$. In this work, we analyze the FNT in the context of a specific family of downward closed sets $A_{m,n,p}$, defined as all multi-indices with $\ell^p$ norm less than $n$ with $p \in [0,\infty]$. These sets induce the downward closed polynomial space $\Pi_{m,n,p}$, within which the FNT algorithm achieves a time complexity of $\mathcal{O}(|A_{m,n,p}|mn)$. We show that this setting, compared to tensor product spaces, yields an improvement in complexity by a factor $\rho_{m,n,p}$, which decays super exponentially with increasing spatial dimension when $m \lesssim n^p$. Additionally, we demonstrate the construction of the hierarchical scheme employed by the FNT and showcase its performance to compute activity scores in sensitivity analysis.