We study additive models built with trend filtering, i.e., additive models
whose components are each regularized by the (discrete) total variation of
their $k$th (discrete) derivative, for a chosen integer $k \geq 0$. This
results in $k$th degree piecewise polynomial components, (e.g., $k=0$ gives
piecewise constant components, $k=1$ gives piecewise linear, $k=2$ gives
piecewise quadratic, etc.). Analogous to its advantages in the univariate case,
additive trend filtering has favorable theoretical and computational
properties, thanks in large part to the localized nature of the (discrete)
total variation regularizer that it uses. On the theory side, we derive fast
error rates for additive trend filtering estimates, and show these rates are
minimax optimal when the underlying function is additive and has component
functions whose derivatives are of bounded variation. We also show that these
rates are unattainable by additive smoothing splines (and by additive models
built from linear smoothers, in general). On the computational side, as per the
standard in additive models, backfitting is an appealing method for
optimization, but it is particularly appealing for additive trend filtering
because we can leverage a few highly efficient univariate trend filtering
solvers. Going one step further, we describe a new backfitting algorithm whose
iterations can be run in parallel, which (as far as we know) is the first of
its kind. Lastly, we present experiments to examine the empirical performance
of additive trend filtering.

本研究采用趋势滤波构建的加法模型，使每个组件都受其 k​​​​​​​​​​​​​(离散) 导数的（离散）全变差正则化，以得到 k​​​​​​​​​​​​​次片段多项式组件。我们推导出加法趋势滤波估计的快速误差率，表明当基础函数为可加性函数且其组件函数的导数具有有界变化时，这些率是极小化最优的。此外，我们描述了一种新的并行迭代的后调优算法，提高了计算效率，并通过实验证明了加法趋势滤波的优越性。