发表机构
Nagoya Mathematical and Information Science Research（名古屋数学与信息科学研究）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究推导Hodrick-Prescott滤波的对偶理论，构建两种趋势表示，提出闭式C³卷积LASSO，经数值验证和基准测试，成功分离NVIDIA股价增长区间断点。
AI 中文摘要
我们推导了Hodrick-Prescott滤波的完整对偶理论，其秩亏的二阶差分惩罚通过广义逆可产生无穷多个等价趋势表示。构建两种典型选择——基于Moore-Penrose的B表示与另一替代的A表示，我们证明提取的趋势在各表示间不变，得到量化共享系数向量下分歧的闭式Bregman型散度，且该散度在λ→∞时消失。我们进一步用紧支撑双权核平滑非光滑的ℓ₁趋势滤波，得到闭式C³卷积LASSO，该方法在不牺牲ℓ₁正则化的拐点检测特性的同时，恢复了牛顿型二次收敛。理论结果经数值验证，并在NVIDIA 2013—2018年每日收盘价上与ADMM/IRLS基准测试，其中稀疏滤波分离出真实增长区间断点——主要是2016年11月财报后加速阶段，与平滑的L₂趋势清晰区分。
英文摘要
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 $λ\to\infty$. We further mollify the non-smooth $\ell_1$ 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 $\ell_1$ 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.
Comments24 pages, 10 figures
