A Generalized Ridge Regression and Convolutional LASSO
Abstract
We derive the complete duality theory underlying the Hodrick--Prescott filter, whose rank-deficient second-difference penalty admits infinitely many equivalent trend representations via generalized inverses. Constructing two canonical choices---the Moore--Penrose-based \emph{B-representation} and an alternative \emph{A-representation}---we prove that the extracted trend is invariant across representations, obtain a closed-form Bregman-type divergence quantifying their disagreement under a shared coefficient vector, and show this divergence vanishes as λ→∞. We further mollify the non-smooth ℓ1 trend filter with a compactly supported biweight kernel to obtain a closed-form C3 \emph{Convolutional LASSO} that restores Newton-type quadratic convergence without sacrificing the kink- setecting character of ℓ1 regularization. Theoretical results are verified numerically and benchmarked against ADMM/IRLS on NVIDIA's 2013--2018 daily closing prices, where the sparse filter isolates genuine growth-regime breaks---chiefly the November 2016 post-earnings acceleration---cleanly separated from the smooth L2 trend.
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BibTeX
@misc{yoshizawa2026generalized,
title = {{A Generalized Ridge Regression and Convolutional LASSO}},
author = {Shintaro Yoshizawa},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.29821},
eprint = {2608.29821},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.29821},
}
RIS
TY - UNPB
TI - A Generalized Ridge Regression and Convolutional LASSO
AU - Shintaro Yoshizawa
PY - 2026
DA - 2026/08/30
PB - arXiv
DO - 10.48550/arxiv.2608.29821
UR - https://arxiv.org/abs/2608.29821
AB - We derive the complete duality theory underlying the Hodrick--Prescott filter, whose rank-deficient second-difference penalty admits infinitely many equivalent trend representations via generalized inverses. Constructing two canonical choices---the Moore--Penrose-based \emph{B-representation} and an alternative \emph{A-representation}---we prove that the extracted trend is invariant across representations, obtain a closed-form Bregman-type divergence quantifying their disagreement under a shared coefficient vector, and show this divergence vanishes as λ→∞. We further mollify the non-smooth ℓ_1 trend filter with a compactly supported biweight kernel to obtain a closed-form C^3 \emph{Convolutional LASSO} that restores Newton-type quadratic convergence without sacrificing the kink- setecting character of ℓ_1 regularization. Theoretical results are verified numerically and benchmarked against ADMM/IRLS on NVIDIA's 2013--2018 daily closing prices, where the sparse filter isolates genuine growth-regime breaks---chiefly the November 2016 post-earnings acceleration---cleanly separated from the smooth L_2 trend.
ER -