% JMLR-ready tables for two-column papers
% Required packages (no siunitx):
% \usepackage{booktabs}
% \usepackage{threeparttable}
% \usepackage{threeparttablex} % for TableNotes + longtable
% \usepackage{longtable}
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\begin{table*}[t]
\centering
\begin{threeparttable}
\caption{Model dimension ranges (min--max across all datasets and folds). Input/Output dimensions follow dataset label spaces.}
\label{tab:model-ranges}
\begin{tabular}{l r r r r r r}
\toprule
Algorithm & Width & Depth & Parameters & Padding & Input dim & Output dim \\
\midrule
AOL & 32--512 & 6--6 & 502--1245037 & 10--524 & 3--262 & 2--100 \\
Orthogonal & 32--512 & 6--6 & 507--1245042 & 10--524 & 3--262 & 2--100 \\
Sandwich & 32--512 & 6--6 & 1057--2620542 & 10--524 & 3--262 & 2--100 \\
SLL & 32--512 & 6--6 & 2326--1622697 & 10--524 & 3--262 & 2--100 \\
LDLT-L & 32--512 & 6--6 & 5480--1454611 & 10--524 & 3--262 & 2--100 \\
LDLT-R & 32--512 & 6--6 & 5577--1588756 & 10--524 & 3--262 & 2--100 \\
\bottomrule
\end{tabular}
\end{threeparttable}
\end{table*}

\begin{table*}[t]
\centering
\begin{threeparttable}
\caption{Sorted mean$\pm$std across $N$ datasets for each algorithm.}
\label{tab:metric_summary}
\begin{tabular}{l r lllll}
\toprule
 &  &  & \multicolumn{4}{c}{Certified Accuracy} \\
\cmidrule(lr){4-7}
Algorithm & $N$ & Accuracy & 36/255 & 72/255 & 108/255 & 255/255 \\
\midrule
AOL & 121 & 0.6049\,\tiny$\pm$0.2396 & 0.2876\,\tiny$\pm$0.3111 & 0.2157\,\tiny$\pm$0.2940 & 0.1739\,\tiny$\pm$0.2716 & 0.0837\,\tiny$\pm$0.1775 \\
Orthogonal & 121 & 0.7036\,\tiny$\pm$0.1911 & 0.6021\,\tiny$\pm$0.2403 & 0.5088\,\tiny$\pm$0.2638 & 0.4229\,\tiny$\pm$0.2741 & 0.1972\,\tiny$\pm$0.2336 \\
Sandwich & 121 & 0.7163\,\tiny$\pm$0.1879 & \textbf{0.6239\,\tiny$\pm$0.2409} & \textbf{0.5483\,\tiny$\pm$0.2616} & \textbf{0.4738\,\tiny$\pm$0.2712} & \textbf{0.2464\,\tiny$\pm$0.2506} \\
SLL & 121 & 0.7011\,\tiny$\pm$0.1939 & 0.5816\,\tiny$\pm$0.2487 & 0.4813\,\tiny$\pm$0.2720 & 0.3961\,\tiny$\pm$0.2799 & 0.1888\,\tiny$\pm$0.2320 \\
\midrule
LDLT-L & 121 & \textbf{0.7245\,\tiny$\pm$0.1908} & 0.4646\,\tiny$\pm$0.3318 & 0.3865\,\tiny$\pm$0.3245 & 0.3247\,\tiny$\pm$0.3088 & 0.1584\,\tiny$\pm$0.2272 \\
LDLT-R & 121 & 0.6970\,\tiny$\pm$0.2021 & 0.6036\,\tiny$\pm$0.2378 & 0.5202\,\tiny$\pm$0.2572 & 0.4478\,\tiny$\pm$0.2640 & 0.2175\,\tiny$\pm$0.2278 \\
\bottomrule
\end{tabular}
\end{threeparttable}
\end{table*}

\begin{table}[t]
\centering
\begin{threeparttable}
{\small
\caption{Overall comparison on Mean Accuracy: average rank (lower is better) with Iman--Davenport $F=34.41$ (df=5,600), $p=1.11e-16$; Nemenyi CD$=0.685$. Counts are significant wins/losses after Holm within-metric at $\alpha=0.05$.}
\label{tab:overall:mean_test_acc}
\setlength{\tabcolsep}{4pt}
\begin{tabular}{@{}l r r r r r r@{}}
\toprule
Algorithm & \shortstack{Avg \\ rank} $\downarrow$ & \shortstack{sig \\ wins} & \shortstack{sig \\ losses} & \shortstack{net \\ wins} & \shortstack{win \\ share} & mean $r$ \\
\midrule
LDLT-L & 2.525 & 4 & 0 & 4 & 0.800 & 0.533 \\
Sandwich & 2.773 & 4 & 0 & 4 & 0.800 & 0.421 \\
Orthogonal & 3.479 & 1 & 2 & -1 & 0.200 & 0.654 \\
SLL & 3.545 & 1 & 2 & -1 & 0.200 & 0.621 \\
LDLT-R & 3.628 & 1 & 2 & -1 & 0.200 & 0.560 \\
AOL & 5.050 & 0 & 5 & -5 & 0.000 & 0.000 \\
\bottomrule
\end{tabular}
}
\end{threeparttable}
\end{table}

\begin{table}[t]
\centering
\begin{threeparttable}
{\small
\caption{Overall comparison on Mean Certified Accuracy (36/255): average rank (lower is better) with Iman--Davenport $F=87.35$ (df=5,600), $p=1.11e-16$; Nemenyi CD$=0.685$. Counts are significant wins/losses after Holm within-metric at $\alpha=0.05$.}
\label{tab:overall:mean_cert_acc_36}
\setlength{\tabcolsep}{4pt}
\begin{tabular}{@{}l r r r r r r@{}}
\toprule
Algorithm & \shortstack{Avg \\ rank} $\downarrow$ & \shortstack{sig \\ wins} & \shortstack{sig \\ losses} & \shortstack{net \\ wins} & \shortstack{win \\ share} & mean $r$ \\
\midrule
Sandwich & 2.211 & 5 & 0 & 5 & 1.000 & 0.539 \\
LDLT-R & 2.603 & 3 & 1 & 2 & 0.600 & 0.599 \\
Orthogonal & 3.202 & 2 & 1 & 1 & 0.400 & 0.693 \\
SLL & 3.289 & 2 & 2 & 0 & 0.400 & 0.704 \\
LDLT-L & 4.091 & 1 & 4 & -3 & 0.200 & 0.648 \\
AOL & 5.603 & 0 & 5 & -5 & 0.000 & 0.000 \\
\bottomrule
\end{tabular}
}
\end{threeparttable}
\end{table}

\begin{table}[t]
\centering
\begin{threeparttable}
{\small
\caption{Overall comparison on Mean Certified Accuracy (72/255): average rank (lower is better) with Iman--Davenport $F=125.56$ (df=5,600), $p=1.11e-16$; Nemenyi CD$=0.685$. Counts are significant wins/losses after Holm within-metric at $\alpha=0.05$.}
\label{tab:overall:mean_cert_acc_72}
\setlength{\tabcolsep}{4pt}
\begin{tabular}{@{}l r r r r r r@{}}
\toprule
Algorithm & \shortstack{Avg \\ rank} $\downarrow$ & \shortstack{sig \\ wins} & \shortstack{sig \\ losses} & \shortstack{net \\ wins} & \shortstack{win \\ share} & mean $r$ \\
\midrule
Sandwich & 1.988 & 5 & 0 & 5 & 1.000 & 0.605 \\
LDLT-R & 2.504 & 4 & 1 & 3 & 0.800 & 0.538 \\
Orthogonal & 3.202 & 3 & 2 & 1 & 0.600 & 0.547 \\
SLL & 3.355 & 2 & 3 & -1 & 0.400 & 0.724 \\
LDLT-L & 4.198 & 1 & 4 & -3 & 0.200 & 0.769 \\
AOL & 5.752 & 0 & 5 & -5 & 0.000 & 0.000 \\
\bottomrule
\end{tabular}
}
\end{threeparttable}
\end{table}

\begin{table}[t]
\centering
\begin{threeparttable}
{\small
\caption{Overall comparison on Mean Certified Accuracy (108/255): average rank (lower is better) with Iman--Davenport $F=134.70$ (df=5,600), $p=1.11e-16$; Nemenyi CD$=0.685$. Counts are significant wins/losses after Holm within-metric at $\alpha=0.05$.}
\label{tab:overall:mean_cert_acc_108}
\setlength{\tabcolsep}{4pt}
\begin{tabular}{@{}l r r r r r r@{}}
\toprule
Algorithm & \shortstack{Avg \\ rank} $\downarrow$ & \shortstack{sig \\ wins} & \shortstack{sig \\ losses} & \shortstack{net \\ wins} & \shortstack{win \\ share} & mean $r$ \\
\midrule
Sandwich & 1.988 & 4 & 0 & 4 & 0.800 & 0.721 \\
LDLT-R & 2.364 & 4 & 0 & 4 & 0.800 & 0.594 \\
Orthogonal & 3.260 & 2 & 2 & 0 & 0.400 & 0.717 \\
SLL & 3.388 & 2 & 2 & 0 & 0.400 & 0.696 \\
LDLT-L & 4.256 & 1 & 4 & -3 & 0.200 & 0.826 \\
AOL & 5.744 & 0 & 5 & -5 & 0.000 & 0.000 \\
\bottomrule
\end{tabular}
}
\end{threeparttable}
\end{table}

\begin{table}[t]
\centering
\begin{threeparttable}
{\small
\caption{Overall comparison on Mean Certified Accuracy (255/255): average rank (lower is better) with Iman--Davenport $F=105.28$ (df=5,600), $p=1.11e-16$; Nemenyi CD$=0.685$. Counts are significant wins/losses after Holm within-metric at $\alpha=0.05$.}
\label{tab:overall:mean_cert_acc_255}
\setlength{\tabcolsep}{4pt}
\begin{tabular}{@{}l r r r r r r@{}}
\toprule
Algorithm & \shortstack{Avg \\ rank} $\downarrow$ & \shortstack{sig \\ wins} & \shortstack{sig \\ losses} & \shortstack{net \\ wins} & \shortstack{win \\ share} & mean $r$ \\
\midrule
Sandwich & 2.000 & 5 & 0 & 5 & 1.000 & 0.643 \\
LDLT-R & 2.467 & 4 & 1 & 3 & 0.800 & 0.620 \\
SLL & 3.293 & 2 & 2 & 0 & 0.400 & 0.664 \\
Orthogonal & 3.442 & 2 & 2 & 0 & 0.400 & 0.652 \\
LDLT-L & 4.227 & 1 & 4 & -3 & 0.200 & 0.846 \\
AOL & 5.570 & 0 & 5 & -5 & 0.000 & 0.000 \\
\bottomrule
\end{tabular}
}
\end{threeparttable}
\end{table}

\begin{table}[t]
\centering
\begin{threeparttable}
{
\caption{Pairwise Wilcoxon outcomes for Mean Accuracy (Holm within-metric at $\alpha=0.05$): row vs. column (\textcolor{green}{$\blacktriangle$} win, \textcolor{red}{$\blacktriangledown$} loss, $\cdot$ none).}
\label{tab:signif:mean_test_acc}
\setlength{\tabcolsep}{3pt}
\begin{tabular}{@{}l c c c c c c @{}}
\toprule
 & AOL & LDLT-L & LDLT-R & Orthogonal & Sandwich & SLL \\
\midrule
AOL & $\cdot$ & \textcolor{red}{$\blacktriangledown$} & \textcolor{red}{$\blacktriangledown$} & \textcolor{red}{$\blacktriangledown$} & \textcolor{red}{$\blacktriangledown$} & \textcolor{red}{$\blacktriangledown$} \\
LDLT-L & \textcolor{green}{$\blacktriangle$} & $\cdot$ & \textcolor{green}{$\blacktriangle$} & \textcolor{green}{$\blacktriangle$} & $\cdot$ & \textcolor{green}{$\blacktriangle$} \\
LDLT-R & \textcolor{green}{$\blacktriangle$} & \textcolor{red}{$\blacktriangledown$} & $\cdot$ & $\cdot$ & \textcolor{red}{$\blacktriangledown$} & $\cdot$ \\
Orthogonal & \textcolor{green}{$\blacktriangle$} & \textcolor{red}{$\blacktriangledown$} & $\cdot$ & $\cdot$ & \textcolor{red}{$\blacktriangledown$} & $\cdot$ \\
Sandwich & \textcolor{green}{$\blacktriangle$} & $\cdot$ & \textcolor{green}{$\blacktriangle$} & \textcolor{green}{$\blacktriangle$} & $\cdot$ & \textcolor{green}{$\blacktriangle$} \\
SLL & \textcolor{green}{$\blacktriangle$} & \textcolor{red}{$\blacktriangledown$} & $\cdot$ & $\cdot$ & \textcolor{red}{$\blacktriangledown$} & $\cdot$ \\
\bottomrule
\end{tabular}
}
\end{threeparttable}
\end{table}

\begin{table}[t]
\centering
\begin{threeparttable}
{
\caption{Pairwise Wilcoxon outcomes for Mean Certified Accuracy (36/255) (Holm within-metric at $\alpha=0.05$): row vs. column (\textcolor{green}{$\blacktriangle$} win, \textcolor{red}{$\blacktriangledown$} loss, $\cdot$ none).}
\label{tab:signif:mean_cert_acc_36}
\setlength{\tabcolsep}{3pt}
\begin{tabular}{@{}l c c c c c c @{}}
\toprule
 & AOL & LDLT-L & LDLT-R & Orthogonal & Sandwich & SLL \\
\midrule
AOL & $\cdot$ & \textcolor{red}{$\blacktriangledown$} & \textcolor{red}{$\blacktriangledown$} & \textcolor{red}{$\blacktriangledown$} & \textcolor{red}{$\blacktriangledown$} & \textcolor{red}{$\blacktriangledown$} \\
LDLT-L & \textcolor{green}{$\blacktriangle$} & $\cdot$ & \textcolor{red}{$\blacktriangledown$} & \textcolor{red}{$\blacktriangledown$} & \textcolor{red}{$\blacktriangledown$} & \textcolor{red}{$\blacktriangledown$} \\
LDLT-R & \textcolor{green}{$\blacktriangle$} & \textcolor{green}{$\blacktriangle$} & $\cdot$ & $\cdot$ & \textcolor{red}{$\blacktriangledown$} & \textcolor{green}{$\blacktriangle$} \\
Orthogonal & \textcolor{green}{$\blacktriangle$} & \textcolor{green}{$\blacktriangle$} & $\cdot$ & $\cdot$ & \textcolor{red}{$\blacktriangledown$} & $\cdot$ \\
Sandwich & \textcolor{green}{$\blacktriangle$} & \textcolor{green}{$\blacktriangle$} & \textcolor{green}{$\blacktriangle$} & \textcolor{green}{$\blacktriangle$} & $\cdot$ & \textcolor{green}{$\blacktriangle$} \\
SLL & \textcolor{green}{$\blacktriangle$} & \textcolor{green}{$\blacktriangle$} & \textcolor{red}{$\blacktriangledown$} & $\cdot$ & \textcolor{red}{$\blacktriangledown$} & $\cdot$ \\
\bottomrule
\end{tabular}
}
\end{threeparttable}
\end{table}

\begin{table}[t]
\centering
\begin{threeparttable}
{
\caption{Pairwise Wilcoxon outcomes for Mean Certified Accuracy (72/255) (Holm within-metric at $\alpha=0.05$): row vs. column (\textcolor{green}{$\blacktriangle$} win, \textcolor{red}{$\blacktriangledown$} loss, $\cdot$ none).}
\label{tab:signif:mean_cert_acc_72}
\setlength{\tabcolsep}{3pt}
\begin{tabular}{@{}l c c c c c c @{}}
\toprule
 & AOL & LDLT-L & LDLT-R & Orthogonal & Sandwich & SLL \\
\midrule
AOL & $\cdot$ & \textcolor{red}{$\blacktriangledown$} & \textcolor{red}{$\blacktriangledown$} & \textcolor{red}{$\blacktriangledown$} & \textcolor{red}{$\blacktriangledown$} & \textcolor{red}{$\blacktriangledown$} \\
LDLT-L & \textcolor{green}{$\blacktriangle$} & $\cdot$ & \textcolor{red}{$\blacktriangledown$} & \textcolor{red}{$\blacktriangledown$} & \textcolor{red}{$\blacktriangledown$} & \textcolor{red}{$\blacktriangledown$} \\
LDLT-R & \textcolor{green}{$\blacktriangle$} & \textcolor{green}{$\blacktriangle$} & $\cdot$ & \textcolor{green}{$\blacktriangle$} & \textcolor{red}{$\blacktriangledown$} & \textcolor{green}{$\blacktriangle$} \\
Orthogonal & \textcolor{green}{$\blacktriangle$} & \textcolor{green}{$\blacktriangle$} & \textcolor{red}{$\blacktriangledown$} & $\cdot$ & \textcolor{red}{$\blacktriangledown$} & \textcolor{green}{$\blacktriangle$} \\
Sandwich & \textcolor{green}{$\blacktriangle$} & \textcolor{green}{$\blacktriangle$} & \textcolor{green}{$\blacktriangle$} & \textcolor{green}{$\blacktriangle$} & $\cdot$ & \textcolor{green}{$\blacktriangle$} \\
SLL & \textcolor{green}{$\blacktriangle$} & \textcolor{green}{$\blacktriangle$} & \textcolor{red}{$\blacktriangledown$} & \textcolor{red}{$\blacktriangledown$} & \textcolor{red}{$\blacktriangledown$} & $\cdot$ \\
\bottomrule
\end{tabular}
}
\end{threeparttable}
\end{table}

\begin{table}[t]
\centering
\begin{threeparttable}
{
\caption{Pairwise Wilcoxon outcomes for Mean Certified Accuracy (108/255) (Holm within-metric at $\alpha=0.05$): row vs. column (\textcolor{green}{$\blacktriangle$} win, \textcolor{red}{$\blacktriangledown$} loss, $\cdot$ none).}
\label{tab:signif:mean_cert_acc_108}
\setlength{\tabcolsep}{3pt}
\begin{tabular}{@{}l c c c c c c @{}}
\toprule
 & AOL & LDLT-L & LDLT-R & Orthogonal & Sandwich & SLL \\
\midrule
AOL & $\cdot$ & \textcolor{red}{$\blacktriangledown$} & \textcolor{red}{$\blacktriangledown$} & \textcolor{red}{$\blacktriangledown$} & \textcolor{red}{$\blacktriangledown$} & \textcolor{red}{$\blacktriangledown$} \\
LDLT-L & \textcolor{green}{$\blacktriangle$} & $\cdot$ & \textcolor{red}{$\blacktriangledown$} & \textcolor{red}{$\blacktriangledown$} & \textcolor{red}{$\blacktriangledown$} & \textcolor{red}{$\blacktriangledown$} \\
LDLT-R & \textcolor{green}{$\blacktriangle$} & \textcolor{green}{$\blacktriangle$} & $\cdot$ & \textcolor{green}{$\blacktriangle$} & $\cdot$ & \textcolor{green}{$\blacktriangle$} \\
Orthogonal & \textcolor{green}{$\blacktriangle$} & \textcolor{green}{$\blacktriangle$} & \textcolor{red}{$\blacktriangledown$} & $\cdot$ & \textcolor{red}{$\blacktriangledown$} & $\cdot$ \\
Sandwich & \textcolor{green}{$\blacktriangle$} & \textcolor{green}{$\blacktriangle$} & $\cdot$ & \textcolor{green}{$\blacktriangle$} & $\cdot$ & \textcolor{green}{$\blacktriangle$} \\
SLL & \textcolor{green}{$\blacktriangle$} & \textcolor{green}{$\blacktriangle$} & \textcolor{red}{$\blacktriangledown$} & $\cdot$ & \textcolor{red}{$\blacktriangledown$} & $\cdot$ \\
\bottomrule
\end{tabular}
}
\end{threeparttable}
\end{table}

\begin{table}[t]
\centering
\begin{threeparttable}
{
\caption{Pairwise Wilcoxon outcomes for Mean Certified Accuracy (255/255) (Holm within-metric at $\alpha=0.05$): row vs. column (\textcolor{green}{$\blacktriangle$} win, \textcolor{red}{$\blacktriangledown$} loss, $\cdot$ none).}
\label{tab:signif:mean_cert_acc_255}
\setlength{\tabcolsep}{3pt}
\begin{tabular}{@{}l c c c c c c @{}}
\toprule
 & AOL & LDLT-L & LDLT-R & Orthogonal & Sandwich & SLL \\
\midrule
AOL & $\cdot$ & \textcolor{red}{$\blacktriangledown$} & \textcolor{red}{$\blacktriangledown$} & \textcolor{red}{$\blacktriangledown$} & \textcolor{red}{$\blacktriangledown$} & \textcolor{red}{$\blacktriangledown$} \\
LDLT-L & \textcolor{green}{$\blacktriangle$} & $\cdot$ & \textcolor{red}{$\blacktriangledown$} & \textcolor{red}{$\blacktriangledown$} & \textcolor{red}{$\blacktriangledown$} & \textcolor{red}{$\blacktriangledown$} \\
LDLT-R & \textcolor{green}{$\blacktriangle$} & \textcolor{green}{$\blacktriangle$} & $\cdot$ & \textcolor{green}{$\blacktriangle$} & \textcolor{red}{$\blacktriangledown$} & \textcolor{green}{$\blacktriangle$} \\
Orthogonal & \textcolor{green}{$\blacktriangle$} & \textcolor{green}{$\blacktriangle$} & \textcolor{red}{$\blacktriangledown$} & $\cdot$ & \textcolor{red}{$\blacktriangledown$} & $\cdot$ \\
Sandwich & \textcolor{green}{$\blacktriangle$} & \textcolor{green}{$\blacktriangle$} & \textcolor{green}{$\blacktriangle$} & \textcolor{green}{$\blacktriangle$} & $\cdot$ & \textcolor{green}{$\blacktriangle$} \\
SLL & \textcolor{green}{$\blacktriangle$} & \textcolor{green}{$\blacktriangle$} & \textcolor{red}{$\blacktriangledown$} & $\cdot$ & \textcolor{red}{$\blacktriangledown$} & $\cdot$ \\
\bottomrule
\end{tabular}
}
\end{threeparttable}
\end{table}


\begin{table*}[t]
\centering
\begin{threeparttable}
\caption[Mean Accuracy]{Wilcoxon signed-rank tests (two-sided) for Mean Accuracy; $p$-values with Holm FWER corrections within-metric and global.}
\label{tab:wilcoxon:mean_test_acc}
\begingroup
\setlength{\tabcolsep}{4pt}
\begin{tabular}{ll r r r r r r r r r r r}
\toprule
\multicolumn{2}{c}{Algorithms} & \multicolumn{6}{c}{Run Statistics} & \multicolumn{5}{c}{Wilcoxon pairwise Statistics} \\\cmidrule(lr){1-2} \cmidrule(lr){3-8} \cmidrule(lr){9-13}
Alg A & Alg B & $n$ & wins$_A$ & wins$_B$ & ties & WinRate A & \shortstack{Median \\ $\Delta$ (A--B)} & $W$ & $p$ & $p_{\text{Holm,within}}$ & $p_{\text{Holm,global}}$ & $r$ \\
\midrule
AOL & LDLT-L & 121 & 16 & 105 & 0 & 0.1322 & -0.0632 & 409 & $2.1e-17^{***}$ & $3.2e-16^{***}$ & $0^{***}$ & 0.7715 \\
AOL & Sandwich & 121 & 23 & 98 & 0 & 0.1901 & -0.0656 & 698 & $0^{***}$ & $0^{***}$ & $0^{***}$ & 0.7036 \\
AOL & Orthogonal & 121 & 25 & 96 & 0 & 0.2066 & -0.0525 & 908 & $0^{***}$ & $0^{***}$ & $0^{***}$ & 0.6542 \\
AOL & SLL & 121 & 24 & 97 & 0 & 0.1983 & -0.0445 & 1047 & $0^{***}$ & $1.0e-10^{***}$ & $3.0e-10^{***}$ & 0.6215 \\
AOL & LDLT-R & 121 & 27 & 94 & 0 & 0.2231 & -0.0357 & 1307 & $7.0e-10^{***}$ & $7.8e-09^{***}$ & $2.3e-08^{***}$ & 0.5604 \\
LDLT-L & SLL & 121 & 86 & 35 & 0 & 0.7107 & 0.0103 & 1511 & $1.7e-08^{***}$ & $1.7e-07^{***}$ & $5.0e-07^{***}$ & 0.5124 \\
LDLT-L & LDLT-R & 121 & 87 & 34 & 0 & 0.7190 & 0.0125 & 1779 & $7.7e-07^{***}$ & $6.9e-06^{***}$ & $2.0e-05^{***}$ & 0.4494 \\
LDLT-L & Orthogonal & 121 & 78 & 43 & 0 & 0.6446 & 0.0104 & 2002 & $1.3e-05^{***}$ & $1.0e-04^{***}$ & $2.5e-04^{***}$ & 0.3969 \\
Orthogonal & Sandwich & 121 & 39 & 81 & 1 & 0.3264 & -0.0079 & 2171 & $1.3e-04^{***}$ & $9.4e-04^{***}$ & $2.5e-03^{**}$ & 0.3487 \\
Sandwich & SLL & 121 & 80 & 41 & 0 & 0.6612 & 0.0063 & 2279 & $2.6e-04^{***}$ & $1.6e-03^{**}$ & $4.6e-03^{**}$ & 0.3318 \\
LDLT-R & Sandwich & 121 & 47 & 74 & 0 & 0.3884 & -0.0101 & 2411 & $9.4e-04^{***}$ & $4.7e-03^{**}$ & $1.4e-02^{*}$ & 0.3008 \\
LDLT-L & Sandwich & 121 & 64 & 56 & 1 & 0.5331 & 0.0008 & 3145 & $2.0e-01$ & $8.2e-01$ & $1.0e+00$ & 0.1158 \\
LDLT-R & Orthogonal & 121 & 54 & 67 & 0 & 0.4463 & -0.0024 & 3206 & $2.1e-01$ & $8.2e-01$ & $1.0e+00$ & 0.1138 \\
Orthogonal & SLL & 121 & 60 & 61 & 0 & 0.4959 & -0.0000 & 3382 & $4.3e-01$ & $8.5e-01$ & $1.0e+00$ & 0.0724 \\
LDLT-R & SLL & 121 & 58 & 63 & 0 & 0.4793 & -0.0010 & 3627 & $8.7e-01$ & $8.7e-01$ & $1.0e+00$ & 0.0148 \\
\bottomrule
\end{tabular}
\begin{tablenotes}
\item Stars mark significance ($^*\,p\!\le\!0.05$, $^{**}\,p\!\le\!0.01$, $^{***}\,p\!\le\!0.001$).
\end{tablenotes}
\endgroup
\end{threeparttable}
\end{table*}



\begin{table*}[t]
\centering
\begin{threeparttable}
\caption[Mean Certified Accuracy (36/255)]{Wilcoxon signed-rank tests (two-sided) for Mean Certified Accuracy (36/255); $p$-values with Holm FWER corrections within-metric and global.}
\label{tab:wilcoxon:mean_cert_acc_36}
\begingroup
\setlength{\tabcolsep}{4pt}
\begin{tabular}{ll r r r r r r r r r r r}
\toprule
\multicolumn{2}{c}{Algorithms} & \multicolumn{6}{c}{Run Statistics} & \multicolumn{5}{c}{Wilcoxon pairwise Statistics} \\\cmidrule(lr){1-2} \cmidrule(lr){3-8} \cmidrule(lr){9-13}
Alg A & Alg B & $n$ & wins$_A$ & wins$_B$ & ties & WinRate A & \shortstack{Median \\ $\Delta$ (A--B)} & $W$ & $p$ & $p_{\text{Holm,within}}$ & $p_{\text{Holm,global}}$ & $r$ \\
\midrule
AOL & LDLT-R & 121 & 3 & 118 & 0 & 0.0248 & -0.2903 & 53 & $5.1e-21^{***}$ & $7.6e-20^{***}$ & $3.6e-19^{***}$ & 0.8552 \\
AOL & SLL & 121 & 4 & 117 & 0 & 0.0331 & -0.2718 & 115 & $2.3e-20^{***}$ & $3.2e-19^{***}$ & $1.5e-18^{***}$ & 0.8406 \\
AOL & Sandwich & 121 & 9 & 112 & 0 & 0.0744 & -0.3414 & 135 & $3.7e-20^{***}$ & $4.9e-19^{***}$ & $2.4e-18^{***}$ & 0.8359 \\
AOL & Orthogonal & 121 & 8 & 113 & 0 & 0.0661 & -0.2982 & 140 & $4.2e-20^{***}$ & $5.1e-19^{***}$ & $2.7e-18^{***}$ & 0.8348 \\
LDLT-L & Sandwich & 121 & 22 & 99 & 0 & 0.1818 & -0.0761 & 770 & $0^{***}$ & $0^{***}$ & $0^{***}$ & 0.6866 \\
AOL & LDLT-L & 121 & 22 & 95 & 4 & 0.1983 & -0.1112 & 872 & $0^{***}$ & $0^{***}$ & $1.0e-10^{***}$ & 0.6485 \\
LDLT-L & LDLT-R & 121 & 34 & 87 & 0 & 0.2810 & -0.0613 & 1073 & $0^{***}$ & $1.0e-10^{***}$ & $5.0e-10^{***}$ & 0.6154 \\
LDLT-L & SLL & 121 & 38 & 83 & 0 & 0.3140 & -0.0545 & 1275 & $4.0e-10^{***}$ & $3.4e-09^{***}$ & $1.4e-08^{***}$ & 0.5679 \\
LDLT-L & Orthogonal & 121 & 40 & 81 & 0 & 0.3306 & -0.0581 & 1346 & $1.3e-09^{***}$ & $9.4e-09^{***}$ & $4.3e-08^{***}$ & 0.5512 \\
Sandwich & SLL & 121 & 89 & 32 & 0 & 0.7355 & 0.0236 & 1459 & $7.9e-09^{***}$ & $4.7e-08^{***}$ & $2.4e-07^{***}$ & 0.5246 \\
Orthogonal & Sandwich & 121 & 33 & 87 & 1 & 0.2769 & -0.0164 & 1846 & $3.0e-06^{***}$ & $1.5e-05^{***}$ & $6.4e-05^{***}$ & 0.4264 \\
LDLT-R & SLL & 121 & 81 & 40 & 0 & 0.6694 & 0.0100 & 2305 & $3.4e-04^{***}$ & $1.4e-03^{**}$ & $5.4e-03^{**}$ & 0.3257 \\
LDLT-R & Sandwich & 121 & 50 & 71 & 0 & 0.4132 & -0.0094 & 2753 & $1.5e-02^{*}$ & $4.6e-02^{*}$ & $1.8e-01$ & 0.2203 \\
Orthogonal & SLL & 121 & 65 & 56 & 0 & 0.5372 & 0.0014 & 2942 & $5.3e-02$ & $1.1e-01$ & $3.9e-01$ & 0.1759 \\
LDLT-R & Orthogonal & 121 & 75 & 46 & 0 & 0.6198 & 0.0065 & 3172 & $1.8e-01$ & $1.8e-01$ & $1.0e+00$ & 0.1218 \\
\bottomrule
\end{tabular}
\begin{tablenotes}
\item Stars mark significance ($^*\,p\!\le\!0.05$, $^{**}\,p\!\le\!0.01$, $^{***}\,p\!\le\!0.001$).
\end{tablenotes}
\endgroup
\end{threeparttable}
\end{table*}



\begin{table*}[t]
\centering
\begin{threeparttable}
\caption[Mean Certified Accuracy (72/255)]{Wilcoxon signed-rank tests (two-sided) for Mean Certified Accuracy (72/255); $p$-values with Holm FWER corrections within-metric and global.}
\label{tab:wilcoxon:mean_cert_acc_72}
\begingroup
\setlength{\tabcolsep}{4pt}
\begin{tabular}{ll r r r r r r r r r r r}
\toprule
\multicolumn{2}{c}{Algorithms} & \multicolumn{6}{c}{Run Statistics} & \multicolumn{5}{c}{Wilcoxon pairwise Statistics} \\\cmidrule(lr){1-2} \cmidrule(lr){3-8} \cmidrule(lr){9-13}
Alg A & Alg B & $n$ & wins$_A$ & wins$_B$ & ties & WinRate A & \shortstack{Median \\ $\Delta$ (A--B)} & $W$ & $p$ & $p_{\text{Holm,within}}$ & $p_{\text{Holm,global}}$ & $r$ \\
\midrule
AOL & SLL & 121 & 0 & 121 & 0 & 0.0000 & -0.2418 & 0 & $1.4e-21^{***}$ & $2.1e-20^{***}$ & $1.0e-19^{***}$ & 0.8677 \\
AOL & LDLT-R & 121 & 1 & 120 & 0 & 0.0083 & -0.2825 & 25 & $2.5e-21^{***}$ & $3.6e-20^{***}$ & $1.9e-19^{***}$ & 0.8618 \\
AOL & Orthogonal & 121 & 4 & 117 & 0 & 0.0331 & -0.2682 & 45 & $4.2e-21^{***}$ & $5.4e-20^{***}$ & $3.0e-19^{***}$ & 0.8571 \\
AOL & Sandwich & 121 & 6 & 115 & 0 & 0.0496 & -0.3329 & 80 & $9.9e-21^{***}$ & $1.2e-19^{***}$ & $6.8e-19^{***}$ & 0.8489 \\
AOL & LDLT-L & 121 & 10 & 93 & 18 & 0.1570 & -0.0849 & 304 & $0^{***}$ & $0^{***}$ & $0^{***}$ & 0.7694 \\
LDLT-L & Sandwich & 121 & 16 & 105 & 0 & 0.1322 & -0.0864 & 509 & $1.9e-16^{***}$ & $0^{***}$ & $0^{***}$ & 0.7480 \\
LDLT-L & LDLT-R & 121 & 30 & 91 & 0 & 0.2479 & -0.0765 & 873 & $0^{***}$ & $0^{***}$ & $0^{***}$ & 0.6624 \\
Orthogonal & Sandwich & 121 & 22 & 98 & 1 & 0.1860 & -0.0318 & 1154 & $1.0e-10^{***}$ & $6.0e-10^{***}$ & $3.5e-09^{***}$ & 0.5918 \\
Sandwich & SLL & 121 & 95 & 26 & 0 & 0.7851 & 0.0391 & 1169 & $1.0e-10^{***}$ & $6.0e-10^{***}$ & $2.8e-09^{***}$ & 0.5928 \\
LDLT-L & SLL & 121 & 34 & 87 & 0 & 0.2810 & -0.0437 & 1219 & $2.0e-10^{***}$ & $1.0e-09^{***}$ & $6.2e-09^{***}$ & 0.5810 \\
LDLT-L & Orthogonal & 121 & 36 & 85 & 0 & 0.2975 & -0.0452 & 1256 & $3.0e-10^{***}$ & $1.5e-09^{***}$ & $1.1e-08^{***}$ & 0.5723 \\
LDLT-R & SLL & 121 & 89 & 32 & 0 & 0.7355 & 0.0202 & 1852 & $2.0e-06^{***}$ & $8.0e-06^{***}$ & $5.0e-05^{***}$ & 0.4322 \\
LDLT-R & Sandwich & 121 & 49 & 72 & 0 & 0.4050 & -0.0133 & 2656 & $7.5e-03^{**}$ & $2.2e-02^{*}$ & $9.7e-02$ & 0.2431 \\
Orthogonal & SLL & 121 & 67 & 54 & 0 & 0.5537 & 0.0046 & 2784 & $1.9e-02^{*}$ & $3.8e-02^{*}$ & $2.1e-01$ & 0.2130 \\
LDLT-R & Orthogonal & 121 & 74 & 47 & 0 & 0.6116 & 0.0089 & 2855 & $3.1e-02^{*}$ & $3.8e-02^{*}$ & $3.1e-01$ & 0.1963 \\
\bottomrule
\end{tabular}
\begin{tablenotes}
\item Stars mark significance ($^*\,p\!\le\!0.05$, $^{**}\,p\!\le\!0.01$, $^{***}\,p\!\le\!0.001$).
\end{tablenotes}
\endgroup
\end{threeparttable}
\end{table*}



\begin{table*}[t]
\centering
\begin{threeparttable}
\caption[Mean Certified Accuracy (108/255)]{Wilcoxon signed-rank tests (two-sided) for Mean Certified Accuracy (108/255); $p$-values with Holm FWER corrections within-metric and global.}
\label{tab:wilcoxon:mean_cert_acc_108}
\begingroup
\setlength{\tabcolsep}{4pt}
\begin{tabular}{ll r r r r r r r r r r r}
\toprule
\multicolumn{2}{c}{Algorithms} & \multicolumn{6}{c}{Run Statistics} & \multicolumn{5}{c}{Wilcoxon pairwise Statistics} \\\cmidrule(lr){1-2} \cmidrule(lr){3-8} \cmidrule(lr){9-13}
Alg A & Alg B & $n$ & wins$_A$ & wins$_B$ & ties & WinRate A & \shortstack{Median \\ $\Delta$ (A--B)} & $W$ & $p$ & $p_{\text{Holm,within}}$ & $p_{\text{Holm,global}}$ & $r$ \\
\midrule
AOL & SLL & 121 & 1 & 116 & 4 & 0.0248 & -0.1721 & 34 & $1.5e-20^{***}$ & $1.8e-19^{***}$ & $1.0e-18^{***}$ & 0.8592 \\
AOL & LDLT-R & 121 & 1 & 120 & 0 & 0.0083 & -0.2318 & 45 & $4.2e-21^{***}$ & $6.3e-20^{***}$ & $3.0e-19^{***}$ & 0.8571 \\
AOL & Orthogonal & 121 & 3 & 117 & 1 & 0.0289 & -0.2099 & 60 & $8.9e-21^{***}$ & $1.2e-19^{***}$ & $6.2e-19^{***}$ & 0.8534 \\
AOL & Sandwich & 121 & 5 & 116 & 0 & 0.0413 & -0.2735 & 82 & $1.0e-20^{***}$ & $1.3e-19^{***}$ & $7.0e-19^{***}$ & 0.8484 \\
AOL & LDLT-L & 121 & 6 & 90 & 25 & 0.1529 & -0.0672 & 113 & $5.8e-16^{***}$ & $0^{***}$ & $0^{***}$ & 0.8259 \\
LDLT-L & Sandwich & 121 & 14 & 107 & 0 & 0.1157 & -0.0819 & 450 & $5.3e-17^{***}$ & $5.8e-16^{***}$ & $0^{***}$ & 0.7619 \\
LDLT-L & LDLT-R & 121 & 25 & 96 & 0 & 0.2066 & -0.0715 & 733 & $0^{***}$ & $0^{***}$ & $0^{***}$ & 0.6953 \\
Orthogonal & Sandwich & 121 & 18 & 102 & 1 & 0.1529 & -0.0376 & 945 & $0^{***}$ & $0^{***}$ & $1.0e-10^{***}$ & 0.6418 \\
Sandwich & SLL & 121 & 93 & 28 & 0 & 0.7686 & 0.0435 & 995 & $0^{***}$ & $0^{***}$ & $1.0e-10^{***}$ & 0.6337 \\
LDLT-L & Orthogonal & 121 & 32 & 88 & 1 & 0.2686 & -0.0352 & 1203 & $2.0e-10^{***}$ & $1.3e-09^{***}$ & $7.7e-09^{***}$ & 0.5801 \\
LDLT-L & SLL & 121 & 35 & 82 & 4 & 0.3058 & -0.0294 & 1335 & $8.7e-09^{***}$ & $4.3e-08^{***}$ & $2.6e-07^{***}$ & 0.5321 \\
LDLT-R & SLL & 121 & 89 & 32 & 0 & 0.7355 & 0.0233 & 1603 & $6.7e-08^{***}$ & $2.7e-07^{***}$ & $1.9e-06^{***}$ & 0.4907 \\
LDLT-R & Orthogonal & 121 & 81 & 40 & 0 & 0.6694 & 0.0182 & 2277 & $2.6e-04^{***}$ & $7.7e-04^{***}$ & $4.6e-03^{**}$ & 0.3323 \\
LDLT-R & Sandwich & 121 & 54 & 67 & 0 & 0.4463 & -0.0097 & 2867 & $3.3e-02^{*}$ & $6.7e-02$ & $3.1e-01$ & 0.1935 \\
Orthogonal & SLL & 121 & 67 & 54 & 0 & 0.5537 & 0.0061 & 2930 & $4.9e-02^{*}$ & $6.7e-02$ & $3.9e-01$ & 0.1787 \\
\bottomrule
\end{tabular}
\begin{tablenotes}
\item Stars mark significance ($^*\,p\!\le\!0.05$, $^{**}\,p\!\le\!0.01$, $^{***}\,p\!\le\!0.001$).
\end{tablenotes}
\endgroup
\end{threeparttable}
\end{table*}



\begin{table*}[t]
\centering
\begin{threeparttable}
\caption[Mean Certified Accuracy (255/255)]{Wilcoxon signed-rank tests (two-sided) for Mean Certified Accuracy (255/255); $p$-values with Holm FWER corrections within-metric and global.}
\label{tab:wilcoxon:mean_cert_acc_255}
\begingroup
\setlength{\tabcolsep}{4pt}
\begin{tabular}{ll r r r r r r r r r r r}
\toprule
\multicolumn{2}{c}{Algorithms} & \multicolumn{6}{c}{Run Statistics} & \multicolumn{5}{c}{Wilcoxon pairwise Statistics} \\\cmidrule(lr){1-2} \cmidrule(lr){3-8} \cmidrule(lr){9-13}
Alg A & Alg B & $n$ & wins$_A$ & wins$_B$ & ties & WinRate A & \shortstack{Median \\ $\Delta$ (A--B)} & $W$ & $p$ & $p_{\text{Holm,within}}$ & $p_{\text{Holm,global}}$ & $r$ \\
\midrule
AOL & LDLT-R & 121 & 0 & 110 & 11 & 0.0455 & -0.0717 & 0 & $8.9e-20^{***}$ & $1.3e-18^{***}$ & $5.6e-18^{***}$ & 0.8678 \\
AOL & SLL & 121 & 2 & 101 & 18 & 0.0909 & -0.0429 & 22 & $2.4e-18^{***}$ & $3.1e-17^{***}$ & $1.5e-16^{***}$ & 0.8608 \\
AOL & LDLT-L & 121 & 3 & 84 & 34 & 0.1653 & -0.0172 & 48 & $0^{***}$ & $0^{***}$ & $0^{***}$ & 0.8465 \\
AOL & Orthogonal & 121 & 2 & 103 & 16 & 0.0826 & -0.0488 & 71 & $4.5e-18^{***}$ & $5.3e-17^{***}$ & $2.7e-16^{***}$ & 0.8458 \\
AOL & Sandwich & 121 & 2 & 112 & 7 & 0.0455 & -0.0886 & 74 & $1.4e-19^{***}$ & $1.9e-18^{***}$ & $8.4e-18^{***}$ & 0.8482 \\
LDLT-L & Sandwich & 121 & 10 & 104 & 7 & 0.1116 & -0.0351 & 415 & $5.9e-16^{***}$ & $0^{***}$ & $0^{***}$ & 0.7579 \\
Orthogonal & Sandwich & 121 & 17 & 97 & 7 & 0.1694 & -0.0273 & 465 & $0^{***}$ & $0^{***}$ & $0^{***}$ & 0.7446 \\
LDLT-L & LDLT-R & 121 & 20 & 90 & 11 & 0.2107 & -0.0296 & 685 & $0^{***}$ & $0^{***}$ & $1.0e-10^{***}$ & 0.6731 \\
Sandwich & SLL & 121 & 84 & 30 & 7 & 0.7231 & 0.0164 & 1043 & $3.0e-10^{***}$ & $1.9e-09^{***}$ & $9.6e-09^{***}$ & 0.5916 \\
LDLT-L & SLL & 121 & 27 & 76 & 18 & 0.2975 & -0.0086 & 1234 & $2.0e-06^{***}$ & $1.0e-05^{***}$ & $5.0e-05^{***}$ & 0.4679 \\
LDLT-L & Orthogonal & 121 & 30 & 74 & 17 & 0.3182 & -0.0039 & 1287 & $2.9e-06^{***}$ & $1.0e-05^{***}$ & $6.4e-05^{***}$ & 0.4587 \\
LDLT-R & SLL & 121 & 80 & 31 & 10 & 0.7025 & 0.0147 & 1338 & $1.9e-07^{***}$ & $1.2e-06^{***}$ & $5.2e-06^{***}$ & 0.4942 \\
LDLT-R & Orthogonal & 121 & 79 & 33 & 9 & 0.6901 & 0.0149 & 1541 & $2.5e-06^{***}$ & $1.0e-05^{***}$ & $5.7e-05^{***}$ & 0.4451 \\
LDLT-R & Sandwich & 121 & 45 & 70 & 6 & 0.3967 & -0.0068 & 2292 & $3.6e-03^{**}$ & $7.2e-03^{**}$ & $5.1e-02$ & 0.2713 \\
Orthogonal & SLL & 121 & 51 & 56 & 14 & 0.4793 & 0.0000 & 2810 & $8.1e-01$ & $8.1e-01$ & $1.0e+00$ & 0.0236 \\
\bottomrule
\end{tabular}
\begin{tablenotes}
\item Stars mark significance ($^*\,p\!\le\!0.05$, $^{**}\,p\!\le\!0.01$, $^{***}\,p\!\le\!0.001$).
\end{tablenotes}
\endgroup
\end{threeparttable}
\end{table*}
