[HELP WANTED] Some LaTeX equations not being rendered using KaTeX

[sunnydigital]

This equation:

$$
\argmin_{\mathcal{G} \in \mathcal{G}_{b}} \argmin_{\mathcal{f}_{\mathcal{G}} \in \mathcal{F}} \argmin_{Z} \sum_{(V_i, V_j) \in \mathcal{E}} b_{i,j} || \mathcal{f}_{\mathcal{G}}(\mathbf{z}_i)-\mathcal{f}_{\mathcal{G}}(\mathbf{z}_{j})||^2
$$

isn't being rendered correctly. It is appearing as:

$$ \argmin_{\mathcal{G} \in \mathcal{G}{b}} \argmin{\mathcal{f}{\mathcal{G}} \in \mathcal{F}} \argmin{Z} \sum_{(V_i, V_j) \in \mathcal{E}} b_{i,j} \mathcal{f}{\mathcal{G}}(\mathbf{z}i)-\mathcal{f}{\mathcal{G}}(\mathbf{z}{j}) $$

(The above is the literal output of the LaTeX on my website).

I am using KaTeX through additions to the extend_head.html file. The changes are as such:

<!-- KaTeX support. Source: https://github.com/adityatelange/hugo-PaperMod/issues/236 -->
<!-- I modified the condition to enable KaTeX. -->
{{ if and .Site.Params.math (ne .Params.math false) }}
<link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/[email protected]/dist/katex.min.css" integrity="sha384-GvrOXuhMATgEsSwCs4smul74iXGOixntILdUW9XmUC6+HX0sLNAK3q71HotJqlAn" crossorigin="anonymous">
<!-- The loading of KaTeX is deferred to speed up page rendering -->
<script defer src="https://cdn.jsdelivr.net/npm/[email protected]/dist/katex.min.js" integrity="sha384-cpW21h6RZv/phavutF+AuVYrr+dA8xD9zs6FwLpaCct6O9ctzYFfFr4dgmgccOTx" crossorigin="anonymous"></script>
<!-- To automatically render math in text elements, include the auto-render extension: -->
<script defer src="https://cdn.jsdelivr.net/npm/[email protected]/dist/contrib/auto-render.min.js" integrity="sha384-+VBxd3r6XgURycqtZ117nYw44OOcIax56Z4dCRWbxyPt0Koah1uHoK0o4+/RRE05" crossorigin="anonymous"
    onload="renderMathInElement(document.body,
    {
        delimiters: [
            {left: '$$', right: '$$', display: true},
            {left: '\\[', right: '\\]', display: true},
            {left: '$', right: '$', display: false},
            {left: '\\(', right: '\\)', display: false}
        ]
    }
);">
</script>
{{ end }}
<!-- End KaTeX support. -->

For reference, the equations:

$$
F_{X}(x)=\operatorname{Pr}(X \leq x)= \begin{cases} 
0 &\text{if } x<0 \\\\ 
\frac{1}{2} &\text{if } 0 \leq x<1 \\\\ 
1 &\text{if } x \geq 1 \end{cases}
$$

$$
\begin{aligned}
P\left(X=x | Y=c_{k}\right) &=P\left(X^{(1)}=x^{(1)}, \cdots, X^{(n)}=x^{(n)} | Y=c_{k}\right) \\\\ 
&=\prod_{j=1}^{n} P\left(X^{(j)}=x^{(j)} | Y=c_{k}\right)
\end{aligned}
$$

$$
\begin{aligned} P\left(X=x | Y=c_{k}\right) &=P\left(X^{(1)}=x^{(1)}, \cdots, X^{(n)}=x^{(n)} | Y=c_{k}\right) \\\\ 
&=\prod_{j=1}^{n} P\left(X^{(j)}=x^{(j)} | Y=c_{k}\right) \end{aligned}
$$

work without a hitch. Any and all help would be greatly appreciated! Thanks!

[sunnydigital]

For reference,

\min_{\mathcal{G} \in G_b}{} \min_{f_{\mathcal{G}} \in \mathcal{F}}{} \min_{Z}{} &\sum_{i=1}^{N} || \mathbf{x}_i - f_G (\mathbf{z}_i) ||^2 + \frac{\lambda}{2} \sum_{(V_i, V_j) \in \mathcal{E}} b_{i,j}  || f_G (\mathbf{z}_i) - f_G (\mathbf{z}_j) ||^2

does not work, but either:

\min_{\mathcal{G} \in G_b}{} \min_{f_{\mathcal{G}} \in \mathcal{F}}{} \min_{Z}{} &\sum_{i=1}^{N} || \mathbf{x}_i - f_G (\mathbf{z}_i) ||^2

and

\frac{\lambda}{2} \sum_{(V_i, V_j) \in \mathcal{E}} b_{i,j}  || f_G (\mathbf{z}_i) - f_G (\mathbf{z}_j) ||^2

work well.