% 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 & 4--4 & 282--694837 & 10--524 & 3--262 & 2--100 \\
Orthogonal & 32--512 & 4--4 & 285--694840 & 10--524 & 3--262 & 2--100 \\
Sandwich & 32--512 & 4--4 & 615--1520140 & 10--524 & 3--262 & 2--100 \\
SLL & 32--512 & 4--4 & 1558--1084073 & 10--524 & 3--262 & 2--100 \\
LDLT-L & 32--512 & 4--4 & 3366--929297 & 10--524 & 3--262 & 2--100 \\
LDLT-R & 32--512 & 4--4 & 3463--1063442 & 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 & 0.6295\,\tiny$\pm$0.2278 & 0.3669\,\tiny$\pm$0.2895 & 0.2660\,\tiny$\pm$0.2953 & 0.2076\,\tiny$\pm$0.2819 & 0.0999\,\tiny$\pm$0.1875 \\
Orthogonal & 121 & 0.6969\,\tiny$\pm$0.1938 & 0.5973\,\tiny$\pm$0.2386 & 0.5073\,\tiny$\pm$0.2617 & 0.4300\,\tiny$\pm$0.2702 & 0.1970\,\tiny$\pm$0.2288 \\
Sandwich & 121 & 0.7215\,\tiny$\pm$0.1871 & \textbf{0.6375\,\tiny$\pm$0.2305} & \textbf{0.5593\,\tiny$\pm$0.2503} & \textbf{0.4836\,\tiny$\pm$0.2659} & \textbf{0.2496\,\tiny$\pm$0.2471} \\
SLL & 121 & 0.6978\,\tiny$\pm$0.1998 & 0.5885\,\tiny$\pm$0.2451 & 0.4975\,\tiny$\pm$0.2649 & 0.4146\,\tiny$\pm$0.2715 & 0.1918\,\tiny$\pm$0.2222 \\
\midrule
LDLT-L & 121 & \textbf{0.7223\,\tiny$\pm$0.1868} & 0.5301\,\tiny$\pm$0.2920 & 0.4293\,\tiny$\pm$0.3049 & 0.3535\,\tiny$\pm$0.3003 & 0.1652\,\tiny$\pm$0.2281 \\
LDLT-R & 121 & 0.7022\,\tiny$\pm$0.1944 & 0.6107\,\tiny$\pm$0.2314 & 0.5292\,\tiny$\pm$0.2525 & 0.4492\,\tiny$\pm$0.2655 & 0.2172\,\tiny$\pm$0.2312 \\
\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 $F=44.33$ (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 & 2.434 & 4 & 0 & 4 & 0.800 & 0.577 \\
Sandwich & 2.566 & 4 & 0 & 4 & 0.800 & 0.517 \\
LDLT-R & 3.438 & 1 & 2 & -1 & 0.200 & 0.629 \\
SLL & 3.624 & 1 & 2 & -1 & 0.200 & 0.678 \\
Orthogonal & 3.831 & 1 & 2 & -1 & 0.200 & 0.639 \\
AOL & 5.107 & 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 $F=89.22$ (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 & 2.021 & 5 & 0 & 5 & 1.000 & 0.626 \\
LDLT-R & 2.715 & 4 & 1 & 3 & 0.800 & 0.514 \\
Orthogonal & 3.417 & 2 & 2 & 0 & 0.400 & 0.627 \\
SLL & 3.426 & 2 & 2 & 0 & 0.400 & 0.602 \\
LDLT-L & 3.785 & 1 & 4 & -3 & 0.200 & 0.732 \\
AOL & 5.636 & 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 $F=101.00$ (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 & 1.926 & 5 & 0 & 5 & 1.000 & 0.670 \\
LDLT-R & 2.628 & 4 & 1 & 3 & 0.800 & 0.557 \\
Orthogonal & 3.376 & 2 & 2 & 0 & 0.400 & 0.643 \\
SLL & 3.409 & 2 & 2 & 0 & 0.400 & 0.640 \\
LDLT-L & 4.058 & 1 & 4 & -3 & 0.200 & 0.732 \\
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 (108/255): average rank (lower is better) with Iman--Davenport $F=125.70$ (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 & 1.868 & 5 & 0 & 5 & 1.000 & 0.675 \\
LDLT-R & 2.562 & 4 & 1 & 3 & 0.800 & 0.548 \\
Orthogonal & 3.347 & 2 & 2 & 0 & 0.400 & 0.674 \\
SLL & 3.360 & 2 & 2 & 0 & 0.400 & 0.660 \\
LDLT-L & 4.136 & 1 & 4 & -3 & 0.200 & 0.801 \\
AOL & 5.727 & 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 $F=105.79$ (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 & 1.909 & 5 & 0 & 5 & 1.000 & 0.671 \\
LDLT-R & 2.550 & 4 & 1 & 3 & 0.800 & 0.575 \\
SLL & 3.310 & 2 & 2 & 0 & 0.400 & 0.662 \\
Orthogonal & 3.471 & 2 & 2 & 0 & 0.400 & 0.607 \\
LDLT-L & 4.202 & 1 & 4 & -3 & 0.200 & 0.796 \\
AOL & 5.558 & 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}{$\blacktriangle$} & $\cdot$ & \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{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}{$\blacktriangledown$} & $\cdot$ \\
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 (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}{$\blacktriangle$} & \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{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 & -0.0568 & 412 & $2.3e-17^{***}$ & $3.4e-16^{***}$ & $0^{***}$ & 0.7708 \\
AOL & Sandwich & 121 & 18 & 103 & 0 & 0.1488 & -0.0595 & 518 & $2.3e-16^{***}$ & $0^{***}$ & $0^{***}$ & 0.7459 \\
AOL & SLL & 121 & 22 & 99 & 0 & 0.1818 & -0.0351 & 806 & $0^{***}$ & $0^{***}$ & $0^{***}$ & 0.6782 \\
AOL & Orthogonal & 121 & 27 & 94 & 0 & 0.2231 & -0.0265 & 971 & $0^{***}$ & $0^{***}$ & $1.0e-10^{***}$ & 0.6394 \\
AOL & LDLT-R & 121 & 25 & 96 & 0 & 0.2066 & -0.0313 & 1013 & $0^{***}$ & $0^{***}$ & $2.0e-10^{***}$ & 0.6295 \\
Orthogonal & Sandwich & 121 & 31 & 89 & 1 & 0.2603 & -0.0126 & 1269 & $6.0e-10^{***}$ & $6.3e-09^{***}$ & $2.2e-08^{***}$ & 0.5643 \\
LDLT-L & Orthogonal & 121 & 94 & 27 & 0 & 0.7769 & 0.0139 & 1309 & $7.0e-10^{***}$ & $6.6e-09^{***}$ & $2.5e-08^{***}$ & 0.5599 \\
LDLT-L & SLL & 121 & 86 & 35 & 0 & 0.7107 & 0.0111 & 1503 & $1.5e-08^{***}$ & $1.2e-07^{***}$ & $4.9e-07^{***}$ & 0.5143 \\
LDLT-L & LDLT-R & 121 & 87 & 33 & 1 & 0.7231 & 0.0108 & 1689 & $3.7e-07^{***}$ & $2.6e-06^{***}$ & $1.1e-05^{***}$ & 0.4639 \\
Sandwich & SLL & 121 & 83 & 38 & 0 & 0.6860 & 0.0090 & 1789 & $8.8e-07^{***}$ & $5.3e-06^{***}$ & $2.3e-05^{***}$ & 0.4470 \\
LDLT-R & Sandwich & 121 & 43 & 78 & 0 & 0.3554 & -0.0068 & 2364 & $6.0e-04^{***}$ & $3.0e-03^{**}$ & $7.2e-03^{**}$ & 0.3118 \\
LDLT-R & SLL & 121 & 66 & 54 & 1 & 0.5496 & 0.0019 & 3070 & $1.4e-01$ & $5.7e-01$ & $8.7e-01$ & 0.1338 \\
LDLT-R & Orthogonal & 121 & 71 & 50 & 0 & 0.5868 & 0.0025 & 3138 & $1.5e-01$ & $5.7e-01$ & $8.7e-01$ & 0.1298 \\
Orthogonal & SLL & 121 & 60 & 61 & 0 & 0.4959 & -0.0004 & 3623 & $8.6e-01$ & $1.0e+00$ & $1.0e+00$ & 0.0158 \\
LDLT-L & Sandwich & 121 & 59 & 62 & 0 & 0.4876 & -0.0002 & 3624 & $8.6e-01$ & $1.0e+00$ & $1.0e+00$ & 0.0155 \\
\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 & 6 & 115 & 0 & 0.0496 & -0.2319 & 84 & $1.1e-20^{***}$ & $1.6e-19^{***}$ & $7.7e-19^{***}$ & 0.8479 \\
AOL & Sandwich & 121 & 6 & 115 & 0 & 0.0496 & -0.2436 & 88 & $1.2e-20^{***}$ & $1.7e-19^{***}$ & $8.4e-19^{***}$ & 0.8470 \\
AOL & SLL & 121 & 6 & 115 & 0 & 0.0496 & -0.1912 & 145 & $4.8e-20^{***}$ & $6.2e-19^{***}$ & $3.2e-18^{***}$ & 0.8336 \\
AOL & Orthogonal & 121 & 10 & 111 & 0 & 0.0826 & -0.2187 & 196 & $1.6e-19^{***}$ & $1.9e-18^{***}$ & $1.0e-17^{***}$ & 0.8216 \\
AOL & LDLT-L & 121 & 16 & 105 & 0 & 0.1322 & -0.1218 & 579 & $8.5e-16^{***}$ & $0^{***}$ & $0^{***}$ & 0.7315 \\
LDLT-L & Sandwich & 121 & 28 & 93 & 0 & 0.2314 & -0.0511 & 884 & $0^{***}$ & $0^{***}$ & $0^{***}$ & 0.6598 \\
Orthogonal & Sandwich & 121 & 24 & 96 & 1 & 0.2025 & -0.0275 & 1002 & $0^{***}$ & $0^{***}$ & $2.0e-10^{***}$ & 0.6282 \\
Sandwich & SLL & 121 & 97 & 24 & 0 & 0.8017 & 0.0199 & 1005 & $0^{***}$ & $0^{***}$ & $2.0e-10^{***}$ & 0.6314 \\
LDLT-L & LDLT-R & 121 & 39 & 82 & 0 & 0.3223 & -0.0372 & 1356 & $1.6e-09^{***}$ & $1.1e-08^{***}$ & $5.2e-08^{***}$ & 0.5488 \\
LDLT-L & Orthogonal & 121 & 47 & 74 & 0 & 0.3884 & -0.0190 & 1855 & $2.1e-06^{***}$ & $1.2e-05^{***}$ & $4.8e-05^{***}$ & 0.4315 \\
LDLT-R & SLL & 121 & 81 & 39 & 1 & 0.6736 & 0.0118 & 2104 & $6.5e-05^{***}$ & $2.5e-04^{***}$ & $1.0e-03^{**}$ & 0.3647 \\
LDLT-L & SLL & 121 & 49 & 72 & 0 & 0.4050 & -0.0075 & 2119 & $4.8e-05^{***}$ & $2.4e-04^{***}$ & $9.2e-04^{***}$ & 0.3694 \\
LDLT-R & Sandwich & 121 & 41 & 80 & 0 & 0.3388 & -0.0168 & 2142 & $6.2e-05^{***}$ & $2.5e-04^{***}$ & $1.0e-03^{**}$ & 0.3640 \\
LDLT-R & Orthogonal & 121 & 78 & 43 & 0 & 0.6446 & 0.0126 & 2429 & $1.1e-03^{**}$ & $2.2e-03^{**}$ & $1.1e-02^{*}$ & 0.2965 \\
Orthogonal & SLL & 121 & 60 & 61 & 0 & 0.4959 & -0.0004 & 3076 & $1.1e-01$ & $1.1e-01$ & $8.7e-01$ & 0.1444 \\
\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 & Sandwich & 121 & 7 & 114 & 0 & 0.0579 & -0.2614 & 107 & $1.9e-20^{***}$ & $2.9e-19^{***}$ & $1.3e-18^{***}$ & 0.8425 \\
AOL & LDLT-R & 121 & 5 & 116 & 0 & 0.0413 & -0.2327 & 116 & $2.4e-20^{***}$ & $3.3e-19^{***}$ & $1.6e-18^{***}$ & 0.8404 \\
AOL & SLL & 121 & 5 & 116 & 0 & 0.0413 & -0.1928 & 162 & $7.1e-20^{***}$ & $9.3e-19^{***}$ & $4.6e-18^{***}$ & 0.8296 \\
AOL & Orthogonal & 121 & 11 & 110 & 0 & 0.0909 & -0.2064 & 214 & $2.4e-19^{***}$ & $2.9e-18^{***}$ & $1.5e-17^{***}$ & 0.8174 \\
AOL & LDLT-L & 121 & 19 & 100 & 2 & 0.1653 & -0.1250 & 557 & $0^{***}$ & $0^{***}$ & $0^{***}$ & 0.7323 \\
Orthogonal & Sandwich & 121 & 19 & 101 & 1 & 0.1612 & -0.0346 & 601 & $0^{***}$ & $0^{***}$ & $0^{***}$ & 0.7240 \\
LDLT-L & Sandwich & 121 & 20 & 101 & 0 & 0.1653 & -0.0772 & 607 & $0^{***}$ & $0^{***}$ & $0^{***}$ & 0.7250 \\
Sandwich & SLL & 121 & 95 & 25 & 1 & 0.7893 & 0.0393 & 801 & $0^{***}$ & $0^{***}$ & $0^{***}$ & 0.6762 \\
LDLT-L & LDLT-R & 121 & 30 & 91 & 0 & 0.2479 & -0.0627 & 930 & $0^{***}$ & $0^{***}$ & $0^{***}$ & 0.6490 \\
LDLT-L & Orthogonal & 121 & 42 & 79 & 0 & 0.3471 & -0.0210 & 1696 & $2.5e-07^{***}$ & $1.5e-06^{***}$ & $7.5e-06^{***}$ & 0.4689 \\
LDLT-L & SLL & 121 & 42 & 79 & 0 & 0.3471 & -0.0213 & 1776 & $7.4e-07^{***}$ & $3.7e-06^{***}$ & $2.0e-05^{***}$ & 0.4501 \\
LDLT-R & SLL & 121 & 83 & 38 & 0 & 0.6860 & 0.0192 & 1938 & $5.9e-06^{***}$ & $2.3e-05^{***}$ & $1.3e-04^{***}$ & 0.4120 \\
LDLT-R & Sandwich & 121 & 40 & 81 & 0 & 0.3306 & -0.0197 & 2062 & $2.5e-05^{***}$ & $7.6e-05^{***}$ & $5.2e-04^{***}$ & 0.3828 \\
LDLT-R & Orthogonal & 121 & 78 & 43 & 0 & 0.6446 & 0.0178 & 2306 & $3.4e-04^{***}$ & $6.9e-04^{***}$ & $4.5e-03^{**}$ & 0.3254 \\
Orthogonal & SLL & 121 & 66 & 55 & 0 & 0.5455 & 0.0041 & 3178 & $1.9e-01$ & $1.9e-01$ & $8.7e-01$ & 0.1204 \\
\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 & LDLT-R & 121 & 1 & 120 & 0 & 0.0083 & -0.2229 & 11 & $1.8e-21^{***}$ & $2.7e-20^{***}$ & $1.3e-19^{***}$ & 0.8651 \\
AOL & SLL & 121 & 4 & 117 & 0 & 0.0331 & -0.1616 & 33 & $3.1e-21^{***}$ & $4.3e-20^{***}$ & $2.3e-19^{***}$ & 0.8599 \\
AOL & Orthogonal & 121 & 6 & 115 & 0 & 0.0496 & -0.1829 & 54 & $5.2e-21^{***}$ & $6.8e-20^{***}$ & $3.8e-19^{***}$ & 0.8550 \\
AOL & Sandwich & 121 & 5 & 116 & 0 & 0.0413 & -0.2530 & 63 & $6.5e-21^{***}$ & $7.8e-20^{***}$ & $4.7e-19^{***}$ & 0.8529 \\
AOL & LDLT-L & 121 & 15 & 102 & 4 & 0.1405 & -0.0795 & 265 & $4.5e-18^{***}$ & $5.0e-17^{***}$ & $2.7e-16^{***}$ & 0.8011 \\
LDLT-L & Sandwich & 121 & 16 & 104 & 1 & 0.1364 & -0.0911 & 469 & $1.3e-16^{***}$ & $0^{***}$ & $0^{***}$ & 0.7556 \\
Orthogonal & Sandwich & 121 & 15 & 106 & 0 & 0.1240 & -0.0411 & 634 & $0^{***}$ & $0^{***}$ & $0^{***}$ & 0.7186 \\
Sandwich & SLL & 121 & 95 & 25 & 1 & 0.7893 & 0.0391 & 769 & $0^{***}$ & $0^{***}$ & $0^{***}$ & 0.6839 \\
LDLT-L & LDLT-R & 121 & 28 & 93 & 0 & 0.2314 & -0.0559 & 854 & $0^{***}$ & $0^{***}$ & $0^{***}$ & 0.6669 \\
LDLT-L & Orthogonal & 121 & 39 & 82 & 0 & 0.3223 & -0.0289 & 1595 & $6.0e-08^{***}$ & $3.6e-07^{***}$ & $1.9e-06^{***}$ & 0.4926 \\
LDLT-L & SLL & 121 & 37 & 82 & 2 & 0.3140 & -0.0231 & 1675 & $5.1e-07^{***}$ & $2.5e-06^{***}$ & $1.4e-05^{***}$ & 0.4605 \\
LDLT-R & SLL & 121 & 83 & 38 & 0 & 0.6860 & 0.0187 & 2059 & $2.5e-05^{***}$ & $9.8e-05^{***}$ & $5.2e-04^{***}$ & 0.3835 \\
LDLT-R & Sandwich & 121 & 43 & 78 & 0 & 0.3554 & -0.0213 & 2138 & $6.0e-05^{***}$ & $1.8e-04^{***}$ & $1.0e-03^{**}$ & 0.3649 \\
LDLT-R & Orthogonal & 121 & 77 & 44 & 0 & 0.6364 & 0.0182 & 2506 & $2.2e-03^{**}$ & $4.4e-03^{**}$ & $2.0e-02^{*}$ & 0.2784 \\
Orthogonal & SLL & 121 & 65 & 56 & 0 & 0.5372 & 0.0038 & 3069 & $1.1e-01$ & $1.1e-01$ & $8.7e-01$ & 0.1460 \\
\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 & 1 & 111 & 9 & 0.0455 & -0.0570 & 10 & $5.4e-20^{***}$ & $8.2e-19^{***}$ & $3.6e-18^{***}$ & 0.8651 \\
AOL & SLL & 121 & 3 & 104 & 14 & 0.0826 & -0.0476 & 23 & $5.3e-19^{***}$ & $6.9e-18^{***}$ & $3.2e-17^{***}$ & 0.8610 \\
AOL & Sandwich & 121 & 2 & 112 & 7 & 0.0455 & -0.0803 & 62 & $9.9e-20^{***}$ & $1.4e-18^{***}$ & $6.3e-18^{***}$ & 0.8514 \\
AOL & Orthogonal & 121 & 5 & 105 & 11 & 0.0868 & -0.0363 & 122 & $2.4e-18^{***}$ & $2.8e-17^{***}$ & $1.4e-16^{***}$ & 0.8331 \\
AOL & LDLT-L & 121 & 8 & 85 & 28 & 0.1818 & -0.0160 & 182 & $0^{***}$ & $0^{***}$ & $0^{***}$ & 0.7958 \\
Orthogonal & Sandwich & 121 & 11 & 103 & 7 & 0.1198 & -0.0313 & 258 & $1.4e-17^{***}$ & $1.5e-16^{***}$ & $8.1e-16^{***}$ & 0.7995 \\
LDLT-L & Sandwich & 121 & 12 & 102 & 7 & 0.1281 & -0.0397 & 342 & $1.1e-16^{***}$ & $0^{***}$ & $0^{***}$ & 0.7772 \\
LDLT-L & LDLT-R & 121 & 23 & 89 & 9 & 0.2273 & -0.0242 & 927 & $1.0e-10^{***}$ & $6.0e-10^{***}$ & $3.0e-09^{***}$ & 0.6135 \\
Sandwich & SLL & 121 & 90 & 25 & 6 & 0.7686 & 0.0240 & 977 & $0^{***}$ & $4.0e-10^{***}$ & $1.7e-09^{***}$ & 0.6135 \\
LDLT-L & SLL & 121 & 27 & 79 & 15 & 0.2851 & -0.0091 & 1323 & $1.9e-06^{***}$ & $9.4e-06^{***}$ & $4.5e-05^{***}$ & 0.4629 \\
LDLT-R & SLL & 121 & 80 & 35 & 6 & 0.6860 & 0.0175 & 1603 & $1.4e-06^{***}$ & $8.1e-06^{***}$ & $3.4e-05^{***}$ & 0.4506 \\
LDLT-L & Orthogonal & 121 & 36 & 75 & 10 & 0.3388 & -0.0085 & 1741 & $5.8e-05^{***}$ & $2.3e-04^{***}$ & $1.0e-03^{**}$ & 0.3816 \\
LDLT-R & Orthogonal & 121 & 77 & 40 & 4 & 0.6529 & 0.0132 & 1982 & $6.5e-05^{***}$ & $2.3e-04^{***}$ & $1.0e-03^{**}$ & 0.3694 \\
LDLT-R & Sandwich & 121 & 45 & 73 & 3 & 0.3843 & -0.0078 & 2234 & $6.1e-04^{***}$ & $1.2e-03^{**}$ & $7.2e-03^{**}$ & 0.3154 \\
Orthogonal & SLL & 121 & 55 & 58 & 8 & 0.4876 & 0.0000 & 3068 & $6.6e-01$ & $6.6e-01$ & $1.0e+00$ & 0.0410 \\
\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*}