A Generalized Ridge Regression and Convolutional LASSO

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 $\lambda \to \mathrm{\infty }$. We further mollify the non-smooth ${\mathrm{\ell }}_1$ 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 ${\mathrm{\ell }}_1$ 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.