Adaptive nonparametric regression with the K-nearest neighbour fused lasso

Varying-coefficients for regional quantile via KNN-based LASSO with applications to health outcome study

Health outcomes, such as body mass index and cholesterol levels, are known to be dependent on age and exhibit varying effects with their associated risk factors. In this paper, we propose a novel framework for dynamic modeling of the associations between health outcomes and risk factors using varying-coefficients (VC) regional quantile regression via K-nearest neighbors (KNN) fused Lasso, which captures the time-varying effects of age. The proposed method has strong theoretical properties, including a tight estimation error bound and the ability to detect exact clustered patterns under certain regularity conditions. To efficiently solve the resulting optimization problem, we develop an alternating direction method of multipliers (ADMM) algorithm. Our empirical results demonstrate the efficacy of the proposed method in capturing the complex age-dependent associations between health outcomes and their risk factors.

Risk bounds for quantile trend filtering

We study quantile trend filtering, a recently proposed method for nonparametric quantile regression, with the goal of generalizing existing risk bounds for the usual trend-filtering estimators that perform mean regression. We study both the penalized and the constrained versions, of order $r \geqslant 1$, of univariate quantile trend filtering. Our results show that both the constrained and the penalized versions of order $r \geqslant 1$ attain the minimax rate up to logarithmic factors, when the $(r-1)$th discrete derivative of the true vector of quantiles belongs to the class of bounded-variation signals. Moreover, we show that if the true vector of quantiles is a discrete spline with a few polynomial pieces, then both versions attain a near-parametric rate of convergence. Corresponding results for the usual trend-filtering estimators are known to hold only when the errors are sub-Gaussian. In contrast, our risk bounds are shown to hold under minimal assumptions on the error variables. In particular, no moment assumptions are needed and our results hold under heavy-tailed errors. Our proof techniques are general, and thus can potentially be used to study other nonparametric quantile regression methods. To illustrate this generality, we employ our proof techniques to obtain new results for multivariate quantile total-variation denoising and high-dimensional quantile linear regression.