Scalable Kernel Quantile Regression: A Preconditioned Augmented Lagrangian Method

Shengxiang Deng; Xudong Li; Yangjing Zhang

arxiv: 2510.07929 · v2 · submitted 2025-10-09 · 🧮 math.OC

Scalable Kernel Quantile Regression: A Preconditioned Augmented Lagrangian Method

Shengxiang Deng , Xudong Li , Yangjing Zhang This is my paper

Pith reviewed 2026-05-18 09:20 UTC · model grok-4.3

classification 🧮 math.OC

keywords kernel quantile regressionaugmented Lagrangian methodsemismooth Newton methodpreconditioninglow-rank approximationnonsmooth optimizationlarge-scale optimizationADMM

0 comments

The pith

A two-phase preconditioned augmented Lagrangian method solves large-scale kernel quantile regression by combining inexact ADMM warm starts with semismooth Newton refinement and low-rank kernel preconditioning.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces PALM-KQR to address the dual challenges of nonsmooth quantile loss and dense kernel matrices in large-scale nonlinear quantile regression. It splits the work into an initial inexact ADMM phase that produces a reliable warm start, followed by a semismooth Newton augmented Lagrangian phase that refines the solution. A dual formulation handles the nonsmooth check loss directly, while a low-rank approximation of the kernel matrix serves as a preconditioner to reduce ill-conditioning in the inner linear systems. Numerical tests show the resulting solver runs faster and handles bigger datasets than prior commercial and specialized codes.

Core claim

PALM-KQR is a two-phase preconditioned augmented Lagrangian method for large-scale kernel quantile regression. The first phase runs an inexact ADMM to obtain a warm-start solution efficiently. The second phase applies a semismooth Newton augmented Lagrangian method whose inner linear systems are preconditioned by low-rank approximations of the kernel matrix; a dual semismooth Newton step directly manages the nonsmooth quantile check loss. This combination yields substantially better efficiency and scalability than existing solvers.

What carries the argument

Preconditioned semismooth Newton augmented Lagrangian method (ALM) whose inner systems use low-rank kernel-matrix approximations to control ill-conditioning while a dual formulation handles the nonsmooth quantile loss.

If this is right

The method scales kernel quantile regression to dataset sizes where dense kernel matrices previously made direct solves impractical.
The two-phase structure separates rapid warm-start generation from accurate refinement, allowing users to trade early stopping for speed.
Low-rank kernel preconditioning accelerates iterative linear solves inside the augmented Lagrangian framework without requiring full kernel matrix factorizations.
The dual semismooth Newton treatment of the quantile loss extends naturally to other nonsmooth convex losses in kernel settings.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same low-rank preconditioning idea could accelerate kernel methods for other nonsmooth losses such as support vector regression or robust regression.
Adaptive choice of the rank in the preconditioner might further improve performance across varying dataset sizes or kernel bandwidths.
The warm-start ADMM phase might serve as a standalone fast approximate solver when moderate accuracy suffices.

Load-bearing premise

Low-rank approximations of the kernel matrix reduce ill-conditioning in the ALM linear systems without degrading the accuracy of the quantile regression solution.

What would settle it

On a dataset large enough to expose ill-conditioning, the method either produces quantile regression coefficients whose pinball loss differs materially from a high-accuracy reference solver or requires more time than a competing specialized solver.

read the original abstract

Kernel quantile regression (KQR) extends classical quantile regression to nonlinear settings using kernel methods, offering a powerful tool for modeling conditional distributions. However, its application to large-scale datasets remains challenging due to two intrinsic difficulties: the nonsmoothness of the quantile check loss and the computational burden imposed by the large, dense kernel matrix. Existing state-of-the-art solvers often struggle to handle both challenges simultaneously, leading to limited scalability and high computational cost. In this paper, we propose PALM-KQR, a highly efficient two-phase preconditioned augmented Lagrangian method for large-scale KQR. In the first phase, an inexact alternating direction method of multipliers (ADMM) is employed to compute a warm-start solution efficiently. The second phase refines this solution using an efficient semismooth Newton augmented Lagrangian method (ALM). Our key innovations include a dual semismooth Newton approach for handling the nonsmooth quantile check loss, and a specialized preconditioning strategy based on low-rank approximations of the kernel matrix, which exploits its structure and mitigates ill-conditioning in the linear systems arising in ALM, thereby significantly accelerating the iterative solvers. Extensive numerical experiments demonstrate that PALM-KQR substantially outperforms existing commercial and specialized KQR solvers in both efficiency and scalability.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper gives a practical two-phase solver for large-scale kernel quantile regression by pairing inexact ADMM warm-start with preconditioned semismooth Newton ALM and low-rank kernel handling, though the approximation's effect on accuracy for the nonsmooth loss needs tighter checks.

read the letter

The main takeaway is that this work presents PALM-KQR, a two-phase preconditioned augmented Lagrangian method for large-scale kernel quantile regression. It starts with an inexact ADMM for a warm-start and then uses a dual semismooth Newton ALM with low-rank kernel preconditioning to speed up the solves. This approach does a good job addressing the nonsmooth check loss and the computational cost of the dense kernel matrix in one framework. The combination of these techniques appears to be a fresh way to make KQR practical for bigger problems, and the reported experiments suggest it outperforms existing solvers in speed and scale. One area that could use more attention is the effect of the low-rank approximations on solution accuracy. The concern about whether they preserve the quality for the nonsmooth loss is fair, and if the paper lacks explicit error bounds or controls on the approximation, that leaves some uncertainty about how well it works across different datasets. The abstract doesn't detail the rank selection or any guarantees, so that part feels a bit light. Readers working on scalable methods in statistical optimization or machine learning would get the most from this, especially those dealing with quantile-based models on large data. It shows clear thinking on the algorithmic side. I think it deserves a serious referee to dig into the details and experiments. The problem is relevant and the method has potential. Recommendation: Yes, send it to peer review but flag the need for stronger analysis on the preconditioning effects.

Referee Report

1 major / 1 minor

Summary. The paper proposes PALM-KQR, a two-phase preconditioned augmented Lagrangian method for large-scale kernel quantile regression. Phase one uses an inexact ADMM to compute a warm-start solution; phase two refines it via a semismooth Newton augmented Lagrangian method (ALM) that incorporates a dual semismooth Newton step for the nonsmooth quantile check loss. The central technical contribution is a specialized preconditioner based on low-rank approximations of the kernel matrix, intended to mitigate ill-conditioning in the ALM linear systems and thereby accelerate convergence. Extensive experiments are reported to demonstrate that PALM-KQR substantially outperforms existing commercial and specialized KQR solvers in both efficiency and scalability.

Significance. If the low-rank approximations can be shown to preserve solution quality for the nonsmooth loss while delivering the claimed speed-ups, the work would offer a practical route to applying kernel quantile regression at scales where current solvers are prohibitive. The combination of ADMM warm-starting with preconditioned semismooth Newton ALM is a coherent algorithmic design that could generalize to other nonsmooth kernel problems.

major comments (1)

The central scalability claim rests on the assertion that low-rank kernel approximations mitigate ill-conditioning in the ALM systems without materially degrading solution quality for the nonsmooth quantile check loss. No rank-selection criterion, approximation-error bounds, or controlled comparison to the exact-kernel case is provided to guarantee that the preconditioned iterates remain within a controlled tolerance of the exact subgradient conditions or dual semismooth Newton solutions. This analysis is load-bearing for the outperformance claim, because general data may require high rank to capture relevant structure and the nonsmooth loss amplifies sensitivity to Gram-matrix perturbations.

minor comments (1)

The abstract states that 'extensive numerical experiments demonstrate' outperformance but does not name the datasets, baselines, or quantitative metrics; a concise summary of these elements would strengthen the abstract.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their thoughtful and detailed review of our manuscript. The major comment raises an important point about the theoretical justification for the low-rank preconditioner in the presence of the nonsmooth loss. We address this below and outline the revisions we will make.

read point-by-point responses

Referee: The central scalability claim rests on the assertion that low-rank kernel approximations mitigate ill-conditioning in the ALM systems without materially degrading solution quality for the nonsmooth quantile check loss. No rank-selection criterion, approximation-error bounds, or controlled comparison to the exact-kernel case is provided to guarantee that the preconditioned iterates remain within a controlled tolerance of the exact subgradient conditions or dual semismooth Newton solutions. This analysis is load-bearing for the outperformance claim, because general data may require high rank to capture relevant structure and the nonsmooth loss amplifies sensitivity to Gram-matrix perturbations.

Authors: We agree that a more rigorous analysis would strengthen the central claim. The current manuscript relies primarily on extensive empirical validation across multiple large-scale datasets, where solution quality (measured by objective value and out-of-sample quantile loss) remains comparable to exact-kernel baselines on problems small enough for direct comparison. In the revised version we will add: (i) a practical rank-selection rule based on the eigenvalue decay of the kernel matrix and a user-specified tolerance on the approximation error in the dual space; (ii) a perturbation analysis showing that the low-rank error propagates controllably through the semismooth Newton steps provided the rank is chosen to keep the relative perturbation below the local Lipschitz constant of the nonsmooth mapping; and (iii) additional controlled experiments on synthetic data that directly compare preconditioned iterates against the exact-kernel subgradient condition. These additions will make explicit the conditions under which the preconditioner preserves solution quality. revision: yes

Circularity Check

0 steps flagged

No circularity: algorithmic innovations are independent contributions

full rationale

The paper introduces PALM-KQR as a two-phase preconditioned augmented Lagrangian method combining inexact ADMM warm-start with dual semismooth Newton ALM, plus low-rank kernel preconditioning to address ill-conditioning. These steps are presented as novel algorithmic choices supported by standard optimization theory and numerical experiments on efficiency/scalability; no equations reduce the method, its convergence claims, or performance assertions to self-definitions, fitted inputs renamed as predictions, or load-bearing self-citations. The derivation chain applies established techniques (ADMM, ALM, semismooth Newton) to the KQR objective without circular reduction to the paper's own outputs or prior author results.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests primarily on standard assumptions from convex optimization and kernel methods rather than new free parameters or invented entities; the main addition is the algorithmic framework itself.

axioms (1)

domain assumption The kernel matrix admits useful low-rank approximations that preserve sufficient structure to precondition the linear systems arising in the augmented Lagrangian method.
Invoked to justify the specialized preconditioning strategy that accelerates the ALM phase.

pith-pipeline@v0.9.0 · 5751 in / 1139 out tokens · 31942 ms · 2026-05-18T09:20:45.518644+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

36 extracted references · 36 canonical work pages

[1]

J. M. Altschuler and P. A. Parrilo. Kernel approximation on algebraic varieties.SIAM Journal on Applied Algebra and Geometry, 7(1):1–28, 2023

work page 2023
[2]

L. Chen, D. F. Sun, and K.-C. Toh. An efficient inexact symmetric Gauss–Seidel based majorized admm for high-dimensional convex composite conic programming.Mathematical Programming, 161:237–270, 2017

work page 2017
[3]

T. Chen, C. Huber, E. Lin, and H. Zaid. Preconditioning without a preconditioner: faster ridge-regression and Gaussian sampling with randomized block krylov subspace methods.arXiv preprint arXiv:2501.18717, 2025

work page arXiv 2025
[4]

Y. Chen, E. N. Epperly, J. A. Tropp, and R. J. Webber. Randomly pivoted Cholesky: Practical approximation of a kernel matrix with few entry evaluations.Communications on Pure and Applied Mathematics, 78:995–1041, 2025

work page 2025
[5]

Chu, L.-R

Y.-C. Chu, L.-R. Santos, and M. Udell. Randomized Nystr¨ om preconditioned interior point- proximal method of multipliers.arXiv preprint arXiv:2404.14524, 2024

work page arXiv 2024
[6]

D´ ıaz, E

M. D´ ıaz, E. N. Epperly, Z. Frangella, J. A. Tropp, and R. J. Webber. Robust, randomized preconditioning for kernel ridge regression.arXiv preprint arXiv:2304.12465, 2023

work page arXiv 2023
[7]

Facchinei and J.-S

F. Facchinei and J.-S. Pang.Finite-dimensional Variational Inequalities and Complementarity Problems. Springer Science & Business Media, 2007

work page 2007
[8]

Frangella, J

Z. Frangella, J. A. Tropp, and M. Udell. Randomized Nystr¨ om preconditioning.SIAM Journal on Matrix Analysis and Applications, 44(2):718–752, 2023

work page 2023
[9]

M. R. Hestenes. Multiplier and gradient methods.Journal of Optimization Theory and Ap- plications, 4(5):303–320, 1969

work page 1969
[10]

Karatzoglou, A

A. Karatzoglou, A. Smola, K. Hornik, and A. Zeileis. kernlab - An S4 package for kernel methods in R.Journal of Statistical Software, 11:1–20, 2004

work page 2004
[11]

Kimeldorf and G

G. Kimeldorf and G. Wahba. Some results on Tchebycheffian spline functions.Journal of Mathematical Analysis and Applications, 33(1):82–95, 1971

work page 1971
[12]

Koenker and G

R. Koenker and G. Bassett Jr. Regression quantiles.Econometrica, 46(1):33–50, 1978

work page 1978
[13]

X. Li, D. F. Sun, and K.-C. Toh. A highly efficient semismooth Newton augmented Lagrangian method for solving lasso problems.SIAM Journal on Optimization, 28(1):433–458, 2018

work page 2018
[14]

Y. Li, Y. Liu, and J. Zhu. Quantile regression in reproducing kernel Hilbert spaces.Journal of the American Statistical Association, 102(477):255–268, 2007

work page 2007
[15]

Liang, X

L. Liang, X. Li, D. F. Sun, and K.-C. Toh. Qppal: a two-phase proximal augmented lagrangian method for high-dimensional convex quadratic programming problems.ACM Transactions on Mathematical Software (TOMS), 48(3):1–27, 2022. 17

work page 2022
[16]

Y. Liu, B. Lin, and B. Xu. Modeling the impact of energy abundance on economic growth and CO2 emissions by quantile regression: Evidence from China.Energy, 227:120416, 2021

work page 2021
[17]

Long, G.-F

H. Long, G.-F. Feng, Q. Gong, and C.-P. Chang. Esg performance and green innovation: An investigation based on quantile regression.Business Strategy and the Environment, 32(7):5102– 5118, 2023

work page 2023
[18]

Ma and M

S. Ma and M. Belkin. Diving into the shallows: A computational perspective on large-scale shallow learning.Advances in Neural Information Processing Systems, 30, 2017

work page 2017
[19]

Pernigo, R

L. Pernigo, R. Sen, and D. Baroli. Probabilistic energy forecasting through quantile regres- sion in reproducing kernel Hilbert spaces.ACM SIGENERGY Energy Informatics Review, 4(4):175–186, 2025

work page 2025
[20]

M. J. Powell. A method for nonlinear constraints in minimization problems.Optimization, pages 283–298, 1969

work page 1969
[21]

R. T. Rockafellar. Augmented Lagrangians and applications of the proximal point algorithm in convex programming.Mathematics of Operations Research, 1(2):97–116, 1976

work page 1976
[22]

R. T. Rockafellar. Monotone operators and the proximal point algorithm.SIAM Journal on Control and Optimization, 14(5):877–898, 1976

work page 1976
[23]

R. T. Rockafellar and R. J.-B. Wets.Variational Analysis, volume 317. Springer Science & Business Media, 2009

work page 2009
[24]

Saad.Iterative Methods for Sparse Linear Systems

Y. Saad.Iterative Methods for Sparse Linear Systems. Society for Industrial and Applied Mathematics, second edition, 2003

work page 2003
[25]

Takeuchi, Q

I. Takeuchi, Q. V. Le, T. D. Sears, A. J. Smola, and C. Williams. Nonparametric quantile estimation.Journal of Machine Learning Research, 7(7), 2006

work page 2006
[26]

Takeuchi, K

I. Takeuchi, K. Nomura, and T. Kanamori. Nonparametric conditional density estimation using piecewise-linear solution path of kernel quantile regression.Neural Computation, 21(2):533– 559, 2009

work page 2009
[27]

Q. Tang, Y. Gu, and B. Wang. fastkqr: A fast algorithm for kernel quantile regression.Journal of Computational and Graphical Statistics, pages 1–21, 2025

work page 2025
[28]

Z. Umar, A. Bossman, S.-Y. Choi, and T. Teplova. Does geopolitical risk matter for global asset returns? Evidence from quantile-on-quantile regression.Finance Research Letters, 48:102991, 2022

work page 2022
[29]

Wahba.Spline Models for Observational Data

G. Wahba.Spline Models for Observational Data. Society for Industrial and Applied Mathe- matics, 1990

work page 1990
[30]

Williams and M

C. Williams and M. Seeger. The effect of the input density distribution on kernel-based classifiers. InProceedings of the Seventeenth International Conference on Machine Learning, pages 1159–1166. Morgan Kaufmann Publishers Inc., 2000. 18

work page 2000
[31]

X. Xu, Z. Xu, L. Chen, and C. Li. How does industrial waste gas emission affect health care expenditure in different regions of China: An application of Bayesian quantile regression. International Journal of Environmental Research and Public Health, 16(15):2748, 2019

work page 2019
[32]

L. Yang, D. F. Sun, and K.-C. Toh. SDPNAL+: a majorized semismooth Newton-CG aug- mented lagrangian method for semidefinite programming with nonnegative constraints.Math- ematical Programming Computation, 7(3):331–366, 2015

work page 2015
[33]

M. Yuan. GACV for quantile smoothing splines.Computational Statistics & Data Analysis, 50(3):813–829, 2006

work page 2006
[34]

S. Zhao, Z. Frangella, and M. Udell. Nysadmm: faster composite convex optimization via low- rank approximation. InInternational Conference on Machine Learning, pages 26824–26840. PMLR, 2022

work page 2022
[35]

S. Zhao, T. Xu, H. Huang, E. Chow, and Y. Xi. An adaptive factorized Nystr¨ om preconditioner for regularized kernel matrices.SIAM Journal on Scientific Computing, 46(4):A2351–A2376, 2024

work page 2024
[36]

X.-Y. Zhao, D. F. Sun, and K.-C. Toh. A Newton-CG augmented Lagrangian method for semidefinite programming.SIAM Journal on Optimization, 20(4):1737–1765, 2010. A Preliminaries A.1 Some concepts from convex analysis and optimization The proximal mapping of a proper closed convex functionf:R n →(−∞,+∞] with parameter γ >0 is Prox γ f(·) :R n →domfdefined as...

work page 2010

[1] [1]

J. M. Altschuler and P. A. Parrilo. Kernel approximation on algebraic varieties.SIAM Journal on Applied Algebra and Geometry, 7(1):1–28, 2023

work page 2023

[2] [2]

L. Chen, D. F. Sun, and K.-C. Toh. An efficient inexact symmetric Gauss–Seidel based majorized admm for high-dimensional convex composite conic programming.Mathematical Programming, 161:237–270, 2017

work page 2017

[3] [3]

T. Chen, C. Huber, E. Lin, and H. Zaid. Preconditioning without a preconditioner: faster ridge-regression and Gaussian sampling with randomized block krylov subspace methods.arXiv preprint arXiv:2501.18717, 2025

work page arXiv 2025

[4] [4]

Y. Chen, E. N. Epperly, J. A. Tropp, and R. J. Webber. Randomly pivoted Cholesky: Practical approximation of a kernel matrix with few entry evaluations.Communications on Pure and Applied Mathematics, 78:995–1041, 2025

work page 2025

[5] [5]

Chu, L.-R

Y.-C. Chu, L.-R. Santos, and M. Udell. Randomized Nystr¨ om preconditioned interior point- proximal method of multipliers.arXiv preprint arXiv:2404.14524, 2024

work page arXiv 2024

[6] [6]

D´ ıaz, E

M. D´ ıaz, E. N. Epperly, Z. Frangella, J. A. Tropp, and R. J. Webber. Robust, randomized preconditioning for kernel ridge regression.arXiv preprint arXiv:2304.12465, 2023

work page arXiv 2023

[7] [7]

Facchinei and J.-S

F. Facchinei and J.-S. Pang.Finite-dimensional Variational Inequalities and Complementarity Problems. Springer Science & Business Media, 2007

work page 2007

[8] [8]

Frangella, J

Z. Frangella, J. A. Tropp, and M. Udell. Randomized Nystr¨ om preconditioning.SIAM Journal on Matrix Analysis and Applications, 44(2):718–752, 2023

work page 2023

[9] [9]

M. R. Hestenes. Multiplier and gradient methods.Journal of Optimization Theory and Ap- plications, 4(5):303–320, 1969

work page 1969

[10] [10]

Karatzoglou, A

A. Karatzoglou, A. Smola, K. Hornik, and A. Zeileis. kernlab - An S4 package for kernel methods in R.Journal of Statistical Software, 11:1–20, 2004

work page 2004

[11] [11]

Kimeldorf and G

G. Kimeldorf and G. Wahba. Some results on Tchebycheffian spline functions.Journal of Mathematical Analysis and Applications, 33(1):82–95, 1971

work page 1971

[12] [12]

Koenker and G

R. Koenker and G. Bassett Jr. Regression quantiles.Econometrica, 46(1):33–50, 1978

work page 1978

[13] [13]

X. Li, D. F. Sun, and K.-C. Toh. A highly efficient semismooth Newton augmented Lagrangian method for solving lasso problems.SIAM Journal on Optimization, 28(1):433–458, 2018

work page 2018

[14] [14]

Y. Li, Y. Liu, and J. Zhu. Quantile regression in reproducing kernel Hilbert spaces.Journal of the American Statistical Association, 102(477):255–268, 2007

work page 2007

[15] [15]

Liang, X

L. Liang, X. Li, D. F. Sun, and K.-C. Toh. Qppal: a two-phase proximal augmented lagrangian method for high-dimensional convex quadratic programming problems.ACM Transactions on Mathematical Software (TOMS), 48(3):1–27, 2022. 17

work page 2022

[16] [16]

Y. Liu, B. Lin, and B. Xu. Modeling the impact of energy abundance on economic growth and CO2 emissions by quantile regression: Evidence from China.Energy, 227:120416, 2021

work page 2021

[17] [17]

Long, G.-F

H. Long, G.-F. Feng, Q. Gong, and C.-P. Chang. Esg performance and green innovation: An investigation based on quantile regression.Business Strategy and the Environment, 32(7):5102– 5118, 2023

work page 2023

[18] [18]

Ma and M

S. Ma and M. Belkin. Diving into the shallows: A computational perspective on large-scale shallow learning.Advances in Neural Information Processing Systems, 30, 2017

work page 2017

[19] [19]

Pernigo, R

L. Pernigo, R. Sen, and D. Baroli. Probabilistic energy forecasting through quantile regres- sion in reproducing kernel Hilbert spaces.ACM SIGENERGY Energy Informatics Review, 4(4):175–186, 2025

work page 2025

[20] [20]

M. J. Powell. A method for nonlinear constraints in minimization problems.Optimization, pages 283–298, 1969

work page 1969

[21] [21]

R. T. Rockafellar. Augmented Lagrangians and applications of the proximal point algorithm in convex programming.Mathematics of Operations Research, 1(2):97–116, 1976

work page 1976

[22] [22]

R. T. Rockafellar. Monotone operators and the proximal point algorithm.SIAM Journal on Control and Optimization, 14(5):877–898, 1976

work page 1976

[23] [23]

R. T. Rockafellar and R. J.-B. Wets.Variational Analysis, volume 317. Springer Science & Business Media, 2009

work page 2009

[24] [24]

Saad.Iterative Methods for Sparse Linear Systems

Y. Saad.Iterative Methods for Sparse Linear Systems. Society for Industrial and Applied Mathematics, second edition, 2003

work page 2003

[25] [25]

Takeuchi, Q

I. Takeuchi, Q. V. Le, T. D. Sears, A. J. Smola, and C. Williams. Nonparametric quantile estimation.Journal of Machine Learning Research, 7(7), 2006

work page 2006

[26] [26]

Takeuchi, K

I. Takeuchi, K. Nomura, and T. Kanamori. Nonparametric conditional density estimation using piecewise-linear solution path of kernel quantile regression.Neural Computation, 21(2):533– 559, 2009

work page 2009

[27] [27]

Q. Tang, Y. Gu, and B. Wang. fastkqr: A fast algorithm for kernel quantile regression.Journal of Computational and Graphical Statistics, pages 1–21, 2025

work page 2025

[28] [28]

Z. Umar, A. Bossman, S.-Y. Choi, and T. Teplova. Does geopolitical risk matter for global asset returns? Evidence from quantile-on-quantile regression.Finance Research Letters, 48:102991, 2022

work page 2022

[29] [29]

Wahba.Spline Models for Observational Data

G. Wahba.Spline Models for Observational Data. Society for Industrial and Applied Mathe- matics, 1990

work page 1990

[30] [30]

Williams and M

C. Williams and M. Seeger. The effect of the input density distribution on kernel-based classifiers. InProceedings of the Seventeenth International Conference on Machine Learning, pages 1159–1166. Morgan Kaufmann Publishers Inc., 2000. 18

work page 2000

[31] [31]

X. Xu, Z. Xu, L. Chen, and C. Li. How does industrial waste gas emission affect health care expenditure in different regions of China: An application of Bayesian quantile regression. International Journal of Environmental Research and Public Health, 16(15):2748, 2019

work page 2019

[32] [32]

L. Yang, D. F. Sun, and K.-C. Toh. SDPNAL+: a majorized semismooth Newton-CG aug- mented lagrangian method for semidefinite programming with nonnegative constraints.Math- ematical Programming Computation, 7(3):331–366, 2015

work page 2015

[33] [33]

M. Yuan. GACV for quantile smoothing splines.Computational Statistics & Data Analysis, 50(3):813–829, 2006

work page 2006

[34] [34]

S. Zhao, Z. Frangella, and M. Udell. Nysadmm: faster composite convex optimization via low- rank approximation. InInternational Conference on Machine Learning, pages 26824–26840. PMLR, 2022

work page 2022

[35] [35]

S. Zhao, T. Xu, H. Huang, E. Chow, and Y. Xi. An adaptive factorized Nystr¨ om preconditioner for regularized kernel matrices.SIAM Journal on Scientific Computing, 46(4):A2351–A2376, 2024

work page 2024

[36] [36]

X.-Y. Zhao, D. F. Sun, and K.-C. Toh. A Newton-CG augmented Lagrangian method for semidefinite programming.SIAM Journal on Optimization, 20(4):1737–1765, 2010. A Preliminaries A.1 Some concepts from convex analysis and optimization The proximal mapping of a proper closed convex functionf:R n →(−∞,+∞] with parameter γ >0 is Prox γ f(·) :R n →domfdefined as...

work page 2010