What's the meaning of "double superscript" in overleaf? - latex

Question

I wrote a wrapped equation in overleaf:
\begin{equation}
\begin{aligned}
\y & = \argmin_{\y'} \sum_{i,j}\W_{ij}||\y'_i - \y'_j||_2^2 \\
& = \argmin_{\y'} \sum_{i,j}(\y'_i - \y'_j)^2\W_{ij} \\
& = \argmin_{\y'} \sum_{i,j}(\y'_i^2 + \y'_j^2 - 2\y'_i\y'_j)\W_{ij} \\
& = \argmin_{\y'} \sum_{i,j}\y'_i^2\W_{ij} + \sum_{i,j}\y'_j^2\W_{ij} - 2\sum_{i,j}\y'_i\y'_j\W_{ij} \\ & = \argmin_{\y'} \sum_{i}\y'_i^2D_{ii} + \sum_{j}\y'_j^2D_{jj} - 2\sum_{i,j}\y'_i\y'_j\W_{ij} \\ & = \argmin_{\y'} 2 (\sum_{i}\y'_i^2D_{ii} - \sum_{i,j}\y'_i\y'_j\W_{ij}) \\
& = \argmin_{\y'} 2 {\y'}^\top \L{\y'} \\
\end{aligned}
\end{equation}
The overleaf shows that there are three "double superscript" error in this code, what is that and how can I fix this (although I can compile it successfully)?
I expect to find the solution to this problem.

Answer 1

A double sub-/superscript error occurs when you have more than one sub-/superscript associated with an element. An example is $x^2^2$ which can be interpreted in two ways: ${x^2}^2$ or $x^{2^2}$. Mathematically they mean the same, but from a typesetting point of view they are interpreted differently:
The first - ${x^2}^2$ - sets both exponents in \scriptstyle (so they're the same size) while the second sets the exponents in staggering (reducing) sizes (first one uses \scriptstyle while the second uses \scriptscriptstyle). This difference in typesetting is rather pushed to the user to fix to avoid confusion or misinterpreting what the result should look like. A similar concept holds for subscripts (like $x_2_2$) and combinations of scripts (like `$x_2^2_2$).
What is hidden in your is the use of a prime (like $y'$) which is similar to $y^\prime$:
$x' = x^\prime \neq x\prime$
As such, $y'^2$ is similar to $y^\prime^2$ which is interpreted as a double superscript, causing the error you see.
The way around it is to appropriately group the elements to denote the way you want it represented in output. In your case, this is most likely ${y'}^2$. I've done that below, with some adjustments to other elements:
\documentclass{article}
\usepackage{mathtools}
% https://tex.stackexchange.com/q/5223/5764
\DeclareMathOperator*{\argmax}{arg\,max}
\DeclareMathOperator*{\argmin}{arg\,min}
\DeclarePairedDelimiter{\norm}{\|}{\|}
\newcommand{\y}{\mathcal{Y}}
\newcommand{\W}{\mathcal{W}}
\begin{document}
\begin{equation}
\begin{aligned}
\y & = \argmin_{\y'} \sum_{i, j} \W_{ij} \norm[\big]{ \y'_i - \y'_j }_2^2 \\
& = \argmin_{\y'} \sum_{i, j} \bigl(\y'_i - \y'_j \bigr)^2 \W_{ij} \\
& = \argmin_{\y'} \sum_{i, j} \bigl( {\y'}_i^2 + {\y'}_j^2 - 2 \y'_i \y'_j \bigr) \W_{ij} \\
& = \argmin_{\y'} \sum_{i, j}{\y'}_i^2 \W_{ij} + \sum_{i, j} {\y'}_j^2 \W_{ij} - 2\sum_{i, j} \y'_i \y'_j \W_{ij} \\
& = \argmin_{\y'} \sum_i {\y'}_i^2 D_{ii} + \sum_j {\y'}_j^2 D_{jj} - 2 \sum_{i, j} \y'_i \y'_j \W_{ij} \\
& = \argmin_{\y'} 2 \biggl( \sum_i {\y'}_i^2 D_{ii} - \sum_{i, j} \y'_i \y'_j \W_{ij} \biggr) \\
& = \argmin_{\y'} 2 {\y'}^\top \L \y'
\end{aligned}
\end{equation}
\end{document}