Jul 2026· Statistical analysis and data mining· 0 citations· 32 references
TL;DR
A robust tensor quantile regression method, in which CANDECOMP/PARAFAC (CP) decomposition is employed for dimension reduction, and an exponential‐type penalty (ETP) is imposed at the element‐wise level to achieve sparse variable selection.
Abstract
Tensor‐valued covariates are increasingly common in multiway measurements, but traditional vector‐valued regression ignores their inherent structure and leads to fragility. In practice, such data are frequently contaminated by heavy‐tailed errors and outliers, and least square estimators lack robustness. In this paper, we propose a robust tensor quantile regression method, in which CANDECOMP/PARAFAC (CP) decomposition is employed for dimension reduction, and an exponential‐type penalty (ETP) is imposed at the element‐wise level to achieve sparse variable selection. The ETP smoothly interpolates between and , reducing bias for large coefficients while preserving computational tractability. We develop an efficient algorithm based on alternating direction method of multipliers (ADMM) framework to solve the ETP‐penalized tensor quantile regression estimator. Theoretically, we address the identifiability of the CP decomposition and establish asymptotic properties, including the estimation consistency and the oracle property. Extensive simulation studies under two representative sparse signal settings show that the proposed method substantially improves signal recovery, estimation accuracy, and predictive performance over quantile regression on vectorized covariates and existing tensor regression methods. An empirical analysis of the Beijing dataset further demonstrates superior predictive performance of the proposed method and reveals pronounced spatial and quantile heterogeneity in the effects of major air pollutants.
We introduce a robust nonparametric regression framework for functional covariates that combines functional principal component analysis (FPCA), marginal copula-scale normalization, bounded-score M-estimation, and multivariate Bernstein smoothing. The proposed procedure reduces the infinite-dimensional functional predictor to a low-dimensional score representation, transforms the retained scores onto the compact unit cube, and estimates a conditional M-functional through a smoothly aggregated system of local estimating equations. This construction is designed to accommodate nonlinear regression structure, heavy-tailed score distributions, and response contamination while limiting the influence of extreme observations. Under suitable regularity and undersmoothing conditions, we establish pointwise and uniform consistency, derive explicit convergence rates, and prove asymptotic normality. The limiting variance contains an explicit Bernstein concentration factor that plays a role analogous to the integrated squared kernel in classical nonparametric regression. The analysis also clarifies the interaction among the projection dimension, the Bernstein resolution, the empirical copula transformation, and the effective local sample size. The finite-sample performance of the method is examined through simulations involving heavy-tailed functional scores, Student-t errors, nonlinear regression effects, and increasing response contamination. The proposed estimator exhibits strong overall predictive performance and good robustness, with particularly favorable behavior under absolute-error criteria.
Wahiba Bouabsa, F. Alshahrani· Mathematics· 0 citations
A distributed stochastic smoothing alternating direction method of multipliers (DSS-ADMM) for horizontally partitioned penalized quantile regression, which characterize the scope of an extension to the minimax concave penalty and the smoothly clipped absolute deviation penalty.
Kernel ridge regression is a standard method for functional data analysis, but its exact behavior is less understood. We study tensor-product kernel ridge regression for estimating the $r$-th moment function of a random function based on noisy discrete observations. The formulation includes mean estimation, covariance estimation, and higher-order moment estimation in a single framework. Our main result gives a precise $1+o_{\mathbb{P}}(1)$ expansion for the $L^2$ error at each admissible regularization parameter. The expansion consists of bias and three variance terms corresponding respectively to variation across the independent sample paths, latent signal variation at each sample point, and variation from measurement errors, identifying the refined error structure underlying functional data. As applications, we show that KRR attains the minimax rate for source smoothness $s \leq 2$ but becomes suboptimal in the sparse regime for $s>2$ due to saturation. A technical ingredient is a set of concentration inequalities for $U$-statistics suited to the dependent product structure of functional observations.
This paper studies trajectory-wise estimation of generalization error for primal--dual algorithms in non-smooth regression. Motivating examples include \(\ell_1\)-penalized least absolute deviations regression and square-root Lasso regression, where the data-fitting loss is non-differentiable and existing risk estimators for gradient-type optimization paths do not apply directly. We develop a general recursive framework that includes the Chambolle--Pock algorithm and related primal--dual splitting methods. We estimate risk by correcting each in-sample fitted value with a weighted combination of past dual iterates. The ideal weights are Stein derivative contractions and depend on the design covariance. We construct replacement weights from observable derivative contractions of the fitted-signal trajectory, yielding a covariance-free, data-driven correction. For high-dimensional Gaussian designs and fixed finite iteration horizon, we prove finite-sample guarantees for both estimators. For square-root ridge, we further establish a matched-Gaussian universality result beyond Gaussian designs. Numerical experiments show that the proposed estimators accurately track the out-of-sample risk along finite optimization paths.
This work proposes a regularized additive tensor autoregressive model with additive interaction of row-wise, column-wise and tube-wise temporal dependence that offers more interpretability, less computational burden due to its convex nature and estimation of the underlying low rank plus sparse pattern of its transition matrices.
D. Ghosh, Nilanjana Chakraborty, S. Roy· 0 citations
This paper proposes a novel transfer learning framework for high‐dimensional quantile regression, addressing the non‐differentiability of quantile loss via a Huber approximation and leveraging elastic net regularization to handle sparse inference. By replacing the piecewise linear quantile check function with a smooth Huber loss, our method achieves computational efficiency while preserving robustness to heavy‐tailed errors and outliers. We develop Oracle Trans‐HAQ, a two‐step transfer algorithm that integrates source knowledge through elastic net penalties, and THAQ, a data‐driven detection framework using cross‐validation to mitigate negative transfer risks in scenarios with unknown informative sources. Numerical simulations demonstrate superior performance in high‐dimensional settings, with significantly lower estimation errors compared to convolution‐smoothed quantile regression and pure quantile loss methods. Applied to GTEx genomic data, our method improves prediction accuracy for JAM2 gene expression quantiles across brain tissues, highlighting its utility in precision medicine for modeling heterogeneous biological effects.
Zixin Lv, Yicheng Liu, Kang Meng et al.· Statistical analysis and dat...· 0 citations
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