% PhD Article 3: Remark on C_D constant for Theorem D (Strang-ISTA for LASSO)
% Created: 2026-03-16 (Feanor worker)
% Status: DRAFT — C_D formula to be verified analytically with Oselodets (April 2026)
%
% Context:
%   F(θ) = ½‖Aθ - y‖² + λ‖θ‖₁
%   LT (proximal gradient):  θ_{k+1} = prox_{hλ‖·‖₁}(θ_k - h∇f(θ_k))
%   Strang-ISTA:             θ_{k+1} = prox_{hλ/2}(prox_{hλ/2}(θ_k) - h∇f(prox_{hλ/2}(θ_k)))
%
% Theorem D (sketch):
%   LT:     F(θ̄_K) - F* ≤ ‖θ_0 - θ*‖²/(2hK)
%   Strang: F(θ̄_K) - F* ≤ ‖θ_0 - θ*‖²/(2hK) + C_D · h² · λ²

% ============================================================
% Insert this \remark block after Theorem D in the LASSO section
% ============================================================

\begin{remark}[Constant $C_D$ and Strang fixed-point shift]
\label{rem:cd_formula}
The constant $C_D$ in Theorem~\ref{thm:strang_ista_convergence} characterizes the algorithmic bias
introduced by Strang splitting applied to the LASSO problem. Numerical estimation
(for $A \in \mathbb{R}^{200 \times 20}$, $\lambda = 0.05$, varying $h \in [0.1/L, 0.5/L]$)
yields
\begin{equation}
  C_D \approx 0.77.
  \label{eq:cd_empirical}
\end{equation}

\noindent \textbf{Analytical conjecture.}
At the Strang fixed point $\bar{\theta}^h$ (i.e., $\bar{\theta}^h = T_{\text{Strang},h}(\bar{\theta}^h)$),
the fixed-point equation analysis suggests
\begin{equation}
  C_D = \frac{\|A \cdot \operatorname{sign}(\theta^*)\|^2}{8},
  \label{eq:cd_formula}
\end{equation}
where $\theta^*$ is the optimal solution and $\operatorname{sign}(\theta^*)$ is its support vector.
This formula gives $C_D = \frac{\|A s^*\|^2}{8}$ where $s^* = \operatorname{sign}(\theta^*)$.
For the numerical setting above: $\|A s^*\|^2 \approx 6.15$, giving $C_D \approx 0.769$,
consistent with the empirical estimate~\eqref{eq:cd_empirical}.

\noindent \textbf{Fixed-point shift.}
The Strang fixed point $\bar{\theta}^h$ is shifted from $\theta^*$ by
\begin{equation}
  \bar{\theta}^h - \theta^* \approx \frac{h\lambda}{2} \operatorname{sign}(\theta^*),
  \label{eq:strang_shift}
\end{equation}
on the support of $\theta^*$.
This bias is proportional to the step size $h$ and regularization strength $\lambda$,
and vanishes in the limit $h \to 0$ (consistent with LT bias = 0).
The corresponding excess objective value is
\begin{equation}
  F(\bar{\theta}^h) - F(\theta^*) = \frac{h^2 \lambda^2}{8} \cdot (s^*)^\top H s^* + O(h^3),
\end{equation}
where $H = A^\top A$ is the Hessian and $s^* = \operatorname{sign}(\theta^*)$.

\noindent \textbf{Empirical verification.}
The estimation script \texttt{article3\_cd\_estimation.py} confirms:
LT (proximal gradient) achieves zero algorithmic bias ($F(\bar{\theta}^h_{\text{LT}}) \approx F^*$
within numerical precision), while Strang-ISTA accumulates a floor of $C_D \cdot h^2 \cdot \lambda^2$
for moderate step sizes $h \in [0.1/L, 0.5/L]$.

\noindent \textit{Remark:} The analytic formula~\eqref{eq:cd_formula} requires a rigorous proof
(planned: discuss with I.~Oseledets, April 2026).
\end{remark}

% ============================================================
% Theorem D placeholder (to be inserted before the remark)
% ============================================================

% \begin{theorem}[Convergence of Strang-ISTA for LASSO]\label{thm:strang_ista_convergence}
% Let $F(\theta) = \frac{1}{2}\|A\theta - y\|^2 + \lambda\|\theta\|_1$ with $A \in \mathbb{R}^{n \times d}$,
% $L = \|A^\top A\|$, and step size $0 < h \leq 1/L$. Define the Strang-ISTA iterates
% $\theta_{k+1} = \mathrm{prox}_{h\lambda/2}(\mathrm{prox}_{h\lambda/2}(\theta_k) - h A^\top(A \mathrm{prox}_{h\lambda/2}(\theta_k) - y))$.
% Then the averaged iterates $\bar{\theta}_K = \frac{1}{K}\sum_{k=1}^{K} \theta_k$ satisfy
% \begin{equation}
%   F(\bar{\theta}_K) - F(\theta^*) \leq \frac{\|\theta_0 - \theta^*\|^2}{2hK} + C_D \cdot h^2 \cdot \lambda^2,
% \end{equation}
% where $C_D \geq 0$ is a constant depending on $A$ and $\theta^*$ (see Remark~\ref{rem:cd_formula}).
% In contrast, standard proximal gradient (LT) achieves $C_D = 0$.
% \end{theorem}