Quantile Additive Trend Filtering

TL;DR

Quantile Additive Trend Filtering (QATF) targets robust estimation in additive models by regularizing each component via the total variation of the $r$-th order discrete derivative under a quantile loss. The authors derive nonasymptotic risk bounds in both fixed and growing dimension regimes, showing a canonical rate of $n^{-2r/(2r+1)}V^{2/(2r+1)}$ (up to log factors) and a dimension-dependent factor when $d$ grows, while remaining robust to heavy-tailed errors. They introduce a practical backfitting algorithm solved via ADMM and validate it on simulated data across diverse error structures and on the World Happiness dataset, demonstrating reliable prediction intervals and interpretable component effects. This work advances theory and practice for robust, interpretable additive models in quantile settings and opens avenues for high-dimensional extensions with sparsity considerations.

Abstract

This paper investigates risk bounds for quantile additive trend filtering, a method gaining increasing significance in the realms of additive trend filtering and quantile regression. We investigate the constrained version of quantile trend filtering within additive models, considering both fixed and growing input dimensions. In the fixed dimension case, we discover an error rate that mirrors the non-quantile minimax rate for additive trend filtering, featuring the main term $n^{-2r/(2r+1)}V^{2/(2r+1)}$, when the underlying quantile function is additive, with components whose $(r-1)$th derivatives are of bounded variation by $V$. In scenarios with a growing input dimension $d$, quantile additive trend filtering introduces a polynomial factor of $d^{(2r+2)/(2r+1)}$. This aligns with the non-quantile variant, featuring a linear factor $d$, particularly pronounced for larger $r$ values. Additionally, we propose a practical algorithm for implementing quantile trend filtering within additive models, using dimension-wise backfitting. We conduct experiments with evenly spaced data points or data that samples from a uniform distribution in the interval $[0,1]$, applying distinct component functions and introducing noise from normal and heavy-tailed distributions. Our findings confirm the estimator's convergence as $n$ increases and its superiority, particularly in heavy-tailed distribution scenarios. These results deepen our understanding of additive trend filtering models in quantile settings, offering valuable insights for practical applications and future research.

Figures (3)

• Figure 1: Component estimates of Quantile Additive Trend Filtering with $\tau = 0.5$ are plotted in blue, and the true component functions $f_{0j}(x) = a_jg_j(x) - b_j$ in black, with $a_j$ and $b_j$ chosen such that $f_{0j}$ has an empirical mean of zero and empirical norm $\|f_{0j}\|_n = 1$. For this scenario, $g_1(x) = \frac{1}{2}x^2$, $g_2(x) = \frac{3}{2} \sin(4 \pi x) + \mathbbm{1}_{x \leq \frac{1}{2}} \cdot \sin(16 \pi x)$, $g_3$ is a dummy dimension (where only 1 randomly assigned point takes non-zero value), and $g_4(x) = e^{3 x} \sin(4 \pi x)$.
• Figure 2: Figure \ref{['fig:3a']} plots true function $f_{0j}(x) = a_jg_j(x) - b_j$ on $0.5$ quantile, where $g_1(x) = \frac{1}{2} \cos(6 \pi x) + 0.1)$ and $g_{2}(x) = -(x - \frac{1}{2})^2$. Figure \ref{['fig:3b']} plots the real data with heavy tailed noise added. Figure \ref{['fig:3c']} plots QATF estimators $\widehat{f}$ on this Scenario, for a reconstruction of the true signal.
• Figure 3: Log-GDP per Capita component fit in Quantile Additive Trend Filtering with $\tau = 0.1, 0.5, \text{and } 0.9$, plotted in purple, black, and orange, respectively.

Theorems & Definitions (62)

• Definition 2.1: Definition 2.1 in sadhanala2019additive
• Definition 2.2: Total Variation
• Theorem 3.1
• Remark 3.2
• Theorem 3.3
• Remark 3.4
• Remark D.2
• Definition E.1
• Definition E.2
• Definition E.3