Draw contour plot for norm penalty term using TikZ

Question

I would like to draw the following contour plots in $\LaTeX$ using the TikZ package.

My script for drawing the contour plots for ridge (q=2) and lasso regression (q=1) are below. The first block of code is ridge (q=2) which is the 2nd plot in the figure above with a circle. The second block of code for lasso (q=1) rotates a rectangle to get the diamond, shown in the 3rd plot.

\usepackage{tikz}

\begin{tikzpicture}
\draw[-] (-0.75,0) -- (0.75,0) node[anchor=north west] {$\beta_1$};
\draw[-] (0,-0.75) -- (0,0.75) node[anchor=south east] {$\beta_2$};
\draw[thick] (0,0) circle (0.5cm);
\end{tikzpicture}

\begin{tikzpicture}
\draw[-] (-0.75,0) -- (0.75,0) node[anchor=north west] {$\beta_1$};
\draw[-] (0,-0.75) -- (0,0.75) node[anchor=south east] {$\beta_2$};
\draw[thick,cm={cos(45) ,-sin(45) ,sin(45) ,cos(45),(0,0)}] (-0.4,-0.4) rectangle (0.4,0.4);
\end{tikzpicture}

With these two completed, I am wondering how to draw the 4th, 5th and 1st plots in the figure above that have non-convex and convex geometric shapes. Finally, I would like to create the following plots for elastic net (q between 1 and 2) using TikZ:

Is there a way to make non-convex (pinched) geometric shapes without resorting to packages outside of TikZ as done in this hyperbola graph? I know there are parabola and arc commands in TikZ, but I think the contour plots require control over how much bending there is towards the corners depending on the value used for q.

Answer 1

This is something that draws cartoons. It has a pic for that, which accepts a parameter. If the parameter is larger than 0, it is more circle-like, and if it is smaller than 0, more cross-like (concave and convex are not uniquely defined here, I think). In addition you can play with the looseness. (You could also combine the axis and norm penalty pics to one, but I feel that this is more restrictive.)

\documentclass[tikz,border=3.14mm]{standalone}
\begin{document}
\begin{tikzpicture}[pics/axis/.style={code={
 \draw[-] (-0.75,0) -- (0.75,0) node[anchor=north west] {$\beta_1$};
 \draw[-] (0,-0.75) -- (0,0.75) node[anchor=south east] {$\beta_2$};
 }},pics/norm penalty/.style={code={
 \draw (0,0.5) to[out=-45+#1,in=135-#1] 
 (0.5,0) to[out=-135+#1,in=45-#1] 
 (0,-0.5) to[out=135+#1,in=-45-#1] 
 (-0.5,0) to[out=45+#1,in=-135-#1]  cycle;}}]
\path (0,0) pic{axis} (0,0) pic[blue,thick,looseness=1.5]{norm penalty=45}
(2,0) pic{axis} (2,0) pic[blue,thick,looseness=1]{norm penalty=45}
(4,0) pic{axis} (4,0) pic[blue,thick,looseness=1]{norm penalty=0}
(6,0) pic{axis} (6,0) pic[blue,thick,looseness=1]{norm penalty=-20}
(8,0) pic{axis} (8,0) pic[blue,thick,looseness=2]{norm penalty=-45}
(2,-2) pic{axis} (2,-2) pic[blue,thick,looseness=0.5]{norm penalty=45}
(4,-2) pic{axis} (4,-2) pic[blue,thick,looseness=0.2]{norm penalty=45};
\end{tikzpicture}
\end{document}

\documentclass[tikz,border=3.14mm]{standalone}
\begin{document}
\begin{tikzpicture}[pics/axis/.style 2 args={code={
 \draw[-] (-0.75,0) -- (0.75,0) node[anchor=north west] {$\beta_1$};
 \draw[-] (0,-0.75) node[below]{#2}-- (0,0.75) node[anchor=south east] {$\beta_2$}
 node[above=0.4cm] {#1};
 }},pics/norm penalty/.style={code={
 \draw (0,0.5) to[out=-45+#1,in=135-#1] 
 (0.5,0) to[out=-135+#1,in=45-#1] 
 (0,-0.5) to[out=135+#1,in=-45-#1] 
 (-0.5,0) to[out=45+#1,in=-135-#1]  cycle;}}]
\path (0,0) pic{axis={$q=1$}{title 1}} (0,0) pic[blue,thick,looseness=1.5]{norm penalty=45}
(2,0) pic{axis={$q=4$}{title 2}} (2,0) pic[blue,thick,looseness=1]{norm penalty=45}
(4,0) pic{axis={$q=2$}{title 3}} (4,0) pic[blue,thick,looseness=1]{norm penalty=0}
(6,0) pic{axis={$q=0.5$}{title 4}} (6,0) pic[blue,thick,looseness=1]{norm penalty=-20}
(8,0) pic{axis={$q=0.1$}{title 5}} (8,0) pic[blue,thick,looseness=2]{norm penalty=-45}
(2,-3) pic{axis={$q=2$}{title 6}} (2,-3) pic[blue,thick,looseness=0.5]{norm penalty=45}
(4,-3) pic{axis={$q=3$}{title 7}} (4,-3) pic[blue,thick,looseness=0.2]{norm penalty=45};
\end{tikzpicture}
\end{document}