From ea0229c32fb0b18c02d1b6ec82fbee132fbd84bd Mon Sep 17 00:00:00 2001
From: ludwigbothmann <46222472+ludwigbothmann@users.noreply.github.com>
Date: Wed, 24 Jan 2024 13:31:34 +0100
Subject: [PATCH] Updates from Overleaf

---
 slides/regularization/slides-regu-l1l2-2.tex | 4 ++--
 1 file changed, 2 insertions(+), 2 deletions(-)

diff --git a/slides/regularization/slides-regu-l1l2-2.tex b/slides/regularization/slides-regu-l1l2-2.tex
index a242935b..d9854306 100644
--- a/slides/regularization/slides-regu-l1l2-2.tex
+++ b/slides/regularization/slides-regu-l1l2-2.tex
@@ -70,12 +70,12 @@
 \end{vbframe}
 
 \begin{vbframe}{$L1$ and $L2$ Reg. with Orthonormal Design}
-For the special case of orthonormal design $\Xmat^{\top}\Xmat=\id$ we can derive closed-form a solution in terms of $\thetah_{\text{OLS}}=(\Xmat^{\top}\Xmat)^{-1}\Xmat^{\top}\yv=\Xmat^{\top}\yv$:
+For special case of orthonormal design $\Xmat^{\top}\Xmat=\id$ we can derive closed-form a solution in terms of $\thetah_{\text{OLS}}=(\Xmat^{\top}\Xmat)^{-1}\Xmat^{\top}\yv=\Xmat^{\top}\yv$:
 $$\thetah_{\text{Lasso}}=\text{sign}(\thetah_{\text{OLS}})(\vert \thetah_{\text{OLS}} \vert - \lambda)_{+}\quad(\text{sparsity})$$
 
 Function $S(\theta,\lambda):=\text{sign}(\theta)(|\theta|-\lambda)_{+}$ is called \textbf{soft thresholding} operator: For $|\theta|<\lambda$ it returns $0$, whereas params $|\theta|>\lambda$ are shrunken toward $0$ by $\lambda$.\\
 \vspace{0.2cm}
-Comparing this to $\thetah_{\text{Ridge}}$ under orthonormal design we see qualitatively different behavior as $\lambda \uparrow$:
+Comparing this to $\thetah_{\text{Ridge}}$ under orthonormal design: %we see qualitatively different behavior as $\lambda \uparrow$:
 $$\thetah_{\text{Ridge}}= ({\Xmat}^T \Xmat  + \lambda \id)^{-1} \Xmat^T\yv=((1+\lambda)\id)^{-1}\thetah_{\text{OLS}} = \frac{\thetah_{\text{OLS}}}{1+\lambda}$$
 
 While soft threshold ensures exact zeros in solution, $L2$ penalty uniformly downscales parameters (no sparsity).