We study a multivariate version of trend filtering, called Kronecker trend filtering or KTF, for the case in which the design points form a lattice in $d$ dimensions. KTF is a natural extension of univariate trend filtering (Steidl et al., 2006; Kim et al., 2009; Tibshirani, 2014), and is defined by minimizing a penalized least squares problem whose penalty term sums the absolute (higher-order) differences of the parameter to be estimated along each of the coordinate directions. The corresponding penalty operator can be written in terms of Kronecker products of univariate trend filtering penalty operators, hence the name Kronecker trend filtering. Equivalently, one can view KTF in terms of an $\ell_1$-penalized basis regression problem where the basis functions are tensor products of falling factorial functions, a piecewise polynomial (discrete spline) basis that underlies univariate trend filtering. This paper is a unification and extension of the results in Sadhanala et al. (2016, 2017). We develop a complete set of theoretical results that describe the behavior of $k^{\mathrm{th}}$ order Kronecker trend filtering in $d$ dimensions, for every $k \geq 0$ and $d \geq 1$. This reveals a number of interesting phenomena, including the dominance of KTF over linear smoothers in estimating heterogeneously smooth functions, and a phase transition at $d=2(k+1)$, a boundary past which (on the high dimension-to-smoothness side) linear smoothers fail to be consistent entirely. We also leverage recent results on discrete splines from Tibshirani (2020), in particular, discrete spline interpolation results that enable us to extend the KTF estimate to any off-lattice location in constant-time (independent of the size of the lattice $n$).

翻译：本文研究多元趋势滤波的一种变体，称为克罗内克趋势滤波（Kronecker trend filtering, KTF），适用于设计点构成$d$维格点的情况。KTF是单变量趋势滤波（Steidl等，2006；Kim等，2009；Tibshirani，2014）的自然推广，其通过最小化惩惩罚最小二乘问题来定义，其中惩罚项沿每个坐标方向对待估计参数的绝对（高阶）差分进行求和。相应的惩罚算子可表示为单变量趋势滤波惩罚算子的克罗内克积，故得名克罗内克趋势滤波。等价地，KTF可视为基于$\ell_1$惩罚的基回归问题，其基函数为下降阶乘函数（一种构成单变量趋势滤波基础的离散样条分片多项式）的张量积。本文统一并推广了Sadhanala等人（2016，2017）的成果。我们建立了一套完整的理论结果，描述了$d$维空间中$k$阶克罗内克趋势滤波的行为（适用于所有$k \geq 0$和$d \geq 1$）。这些结果揭示了若干有趣现象，包括KTF在估计异质平滑函数时优于线性平滑器，以及在$d=2(k+1)$处发生相变——超过这一边界（即高维度-平滑度侧）时，线性平滑器将完全丧失一致性。我们还利用了Tibshirani（2020）关于离散样条的最新成果，特别是离散样条插值结果，使我们能够以恒定时间（与格点规模$n$无关）将KTF估计扩展至任意非格点位置。