\section{Supplementary Experiments}
\label{sec:ap5}



\subsection{Robustness to Model Choice Experiments}
\label{sec:ap5.1}


Here, we present supplementary experiments aimed at further validating the efficacy of our proposed JSSL method. These experiments involve a comparative analysis between JSSL and traditional SSL MRI Reconstruction. We adapt the methodologies outlined in \Section{sec3} of our main paper, utilizing two distinct deep MRI reconstruction models instead of the vSHARP architecture:

\begin{itemize}
    \item Utilizing a plain image domain U-Net \cite{Ronneberger2015}, a non-physics-based model that takes an undersampled-reconstructed image as input and refines it. Specifically, we employ a U-Net with four scales and 64 filters in the first channel.
    
    \item Employing an End-to-end Variational Network (E2EVarNet) \cite{Sriram2020}, a physics-based model that executes a gradient descent-like optimization scheme in the $k$-space domain. For E2EVarNet, we perform 6 optimization steps using U-Nets with four scales and 16 filters in the first scale.
\end{itemize}

To estimate sensitivity maps for both architectures, an identical Sensitivity Map Estimation (SME) module was integrated, mirroring the experimental setup outlined in our primary paper.

Both models underwent training and evaluation on data subsampled with acceleration factors of 4, 8, and 16, with ACS ratios of 8\%, 4\%, and 2\% of the data shape.  Choices of hyperparameters for JSSL and SSL are the same as in the comparative experiments presented in \Section{subsec3.6}. Additionally, choices for proxy and target datasets, as well as data splits, are also the same as in the main paper.


Experimental setups were executed on NVIDIA A100 80GB GPUs, utilizing 2 GPUs for U-Net and 1 GPU for E2EVarNet. We employed batch sizes of 2 and 4 for U-Net and E2EVarNet, respectively, on each GPU. The optimization procedures, initial learning rates, and the employed optimizers aligned with those utilized in the main paper. 

\input{tab2}

\Table{S2} details the model specifics for all considered architectures presented in both the main paper and this section.


\subsubsection{Robustness to Model Choice Experiments Results}

% ------------------------------TABLE-------------------------------------------%
\begin{table*}[!htb]
\centering
\caption{Robustness to model choice experiments results. An asterisk ($*$) denotes that the average best method (bold) was not found to be statistically significantly better than the corresponding method ($p>0.05$).
}
\label{tab:comparison-unet-varnet}
\setlength{\tabcolsep}{0.8pt}
{\renewcommand{\arraystretch}{1.8}
\resizebox{1\textwidth}{!}{%
\begin{tabular}{ccccccccccc}
\hline
\multirow{2}{*}{\textbf{Architecture}} &
  \multirow{2}{*}{\textbf{Setup}} &
  \multicolumn{3}{c}{\textbf{4x}} &
  \multicolumn{3}{c}{\textbf{8x}} &
  \multicolumn{3}{c}{\textbf{16x}} \\ \cline{3-11} 
 &
   &
  SSIM &
  pSNR &
  NMSE &
  SSIM &
  pSNR &
  NMSE &
  SSIM &
  pSNR &
  NMSE \\ \hline
\multirow{2}{*}{U-Net} &
  SSL &
  0.854${\scriptscriptstyle\pm 0.031}$ &
  33.0${\scriptscriptstyle\pm 1.6}$ &
  0.013${\scriptscriptstyle\pm 0.004}$ &
  0.742${\scriptscriptstyle\pm 0.040}$ &
  29.4${\scriptscriptstyle\pm 1.4}$ &
  0.030${\scriptscriptstyle\pm 0.006}$ &
  0.651${\scriptscriptstyle\pm 0.051}$ &
  \textbf{26.7${\scriptscriptstyle\pm 1.5}$} &
  \textbf{0.055${\scriptscriptstyle\pm 0.009}$} \\
 &
  JSSL &
  \textbf{0.863${\scriptscriptstyle\pm 0.031}$} &
  \textbf{33.5${\scriptscriptstyle\pm 1.5}$} &
  \textbf{0.012${\scriptscriptstyle\pm 0.002}$} &
  \textbf{0.759${\scriptscriptstyle\pm 0.042}$} &
  \textbf{29.7${\scriptscriptstyle\pm 1.4}$} &
  \textbf{0.027${\scriptscriptstyle\pm 0.005}$} &
  \textbf{0.663${\scriptscriptstyle\pm 0.051}$} &
  26.7${\scriptscriptstyle\pm 1.4}$$^*$ &
  0.055${\scriptscriptstyle\pm 0.009}$$^*$ \\ \hline
\multirow{2}{*}{E2EVarNet} &
  SSL &
  0.874${\scriptscriptstyle\pm 0.029}$ &
  33.7${\scriptscriptstyle\pm 1.7}$ &
  0.011${\scriptscriptstyle\pm 0.003}$ &
  0.770${\scriptscriptstyle\pm 0.039}$ &
  30.0${\scriptscriptstyle\pm 1.4}$ &
  0.025${\scriptscriptstyle\pm 0.006}$ &
  0.670${\scriptscriptstyle\pm 0.051}$ &
  27.0${\scriptscriptstyle\pm 1.5}$ &
  0.051${\scriptscriptstyle\pm 0.009}$ \\
 &
  JSSL &
  \textbf{0.888${\scriptscriptstyle\pm 0.032}$} &
  \textbf{34.9${\scriptscriptstyle\pm 1.6}$} &
  \textbf{0.008${\scriptscriptstyle\pm 0.002}$} &
  \textbf{0.784${\scriptscriptstyle\pm 0.042}$} &
  \textbf{30.5${\scriptscriptstyle\pm 1.4}$} &
  \textbf{0.023${\scriptscriptstyle\pm 0.005}$} &
  \textbf{0.678${\scriptscriptstyle\pm 0.053}$} &
  \textbf{27.1${\scriptscriptstyle\pm 1.5}$} &
  \textbf{0.050${\scriptscriptstyle\pm 0.009}$} \\ \hline
\end{tabular}%
}
}
\end{table*}
% ------------------------------TABLE-------------------------------------------%

The average results of our supplementary comparative studies to assess JSSL's robustness to different architecture choices are provided  in \Table{comparison-unet-varnet}. From these results we observe alignment with our original findings: JSSL-trained models consistently outperform SSL-trained models for both architecture choices.

Furthermore, the superior performance of vSHARP and E2EVarNet compared to the U-Net model in both SSL and JSSL settings across all acceleration factors highlights the advantage of adopting physics-guided unrolled models for reconstruction. It is also worth mentioning that vSHARP consistently outperformed E2EVarNet at all accelerations.
% \input{fig2}
% \input{fig3}


% \subsection{Alternative Configuration Experiments}
% \label{sec:ap5.2}

% To investigate JSSL further and evaluate its performance under different settings, we examine additional configurations for the JSSL and SSL setups. In particular, we perform the following experiments on the first set of experiments (prostate data as target):
% \begin{enumerate}[label=(\arabic*)]
%     \item JSSL and SSL in all domains by oversampling $10$-fold the target dataset during training to balance better proxy and target data, in comparison to $2$-fold in our original experiments in \Section{subsec3.4}.
%     \item JSSL and SSL using a constant partitioning ratio of $q=0.5$ instead of $q\in(0.3-0.8)$ as in  \Section{subsec3.4}.
%     \item JSSL and SSL setting for the ACS window $w\times w=10 \times 10$ opposed to $w\times w=4\times 4$ in our experiments in \Section{subsec3.4}.
% \end{enumerate}


% \subsubsection{Alternative Configuration Experiments Results}


% % ------------------------------TABLE-------------------------------------------%
% \begin{table*}[!hbt]
% \centering
% \caption{Average results for alternative configurations with the fastMRI prostate as target. 
% % An asterisk ($*$) denotes that the average best method (bold) was not found to be statistically significantly better than the corresponding method ($p>0.05$).
% }
% \resizebox{1\textwidth}{!}{%
% \setlength{\tabcolsep}{0.8pt}
% {\renewcommand{\arraystretch}{1.5}
% \begin{tabular}{ccccccccccccc}
% \hline
% \multirow{2}{*}{\textbf{Setup}} &
%   \multicolumn{3}{c}{\textbf{2x}} &
%   \multicolumn{3}{c}{\textbf{4x}} &
%   \multicolumn{3}{c}{\textbf{8x}} &
%   \multicolumn{3}{c}{\textbf{16x}} \\ \cline{2-13} 
%  &
%   SSIM &
%   pSNR &
%   NMSE &
%   SSIM &
%   pSNR &
%   NMSE &
%   SSIM &
%   pSNR &
%   NMSE &
%   SSIM &
%   pSNR &
%   NMSE \\ \hline
% SSL Original &
%   0.956${\scriptscriptstyle\pm 0.015}$ &
%   38.8${\scriptscriptstyle\pm 2.6}$ &
%   0.004${\scriptscriptstyle\pm 0.002}$ &
%   0.891${\scriptscriptstyle\pm 0.030}$ &
%   34.7${\scriptscriptstyle\pm 2.0}$ &
%   0.009${\scriptscriptstyle\pm 0.003}$ &
%   0.801${\scriptscriptstyle\pm 0.038}$ &
%   31.1${\scriptscriptstyle\pm 1.5}$ &
%   0.020${\scriptscriptstyle\pm 0.005}$ &
%   0.707${\scriptscriptstyle\pm 0.050}$ &
%   28.0${\scriptscriptstyle\pm 1.6}$ &
%   0.041${\scriptscriptstyle\pm 0.008}$ \\
% SSL ALL Original &
%   0.953${\scriptscriptstyle\pm 0.016}$ &
%   38.6${\scriptscriptstyle\pm 2.5}$ &
%   0.004${\scriptscriptstyle\pm 0.002}$ &
%   0.892${\scriptscriptstyle\pm 0.031}$ &
%   34.8${\scriptscriptstyle\pm 2.0}$ &
%   0.009${\scriptscriptstyle\pm 0.004}$ &
%   0.801${\scriptscriptstyle\pm 0.041}$ &
%   31.1${\scriptscriptstyle\pm 1.6}$ &
%   0.020${\scriptscriptstyle\pm 0.006}$ &
%   0.699${\scriptscriptstyle\pm 0.052}$ &
%   27.8${\scriptscriptstyle\pm 1.6}$ &
%   0.043${\scriptscriptstyle\pm 0.010}$ \\
% JSSL Original &
%   \textbf{0.965${\scriptscriptstyle\pm 0.015}$} &
%   \textbf{39.5${\scriptscriptstyle\pm 2.8}$} &
%   \textbf{0.003${\scriptscriptstyle\pm 0.002}$} &
%   \textbf{0.918${\scriptscriptstyle\pm 0.026}$} &
%   \textbf{36.4${\scriptscriptstyle\pm 1.9}$} &
%   \textbf{0.006${\scriptscriptstyle\pm 0.002}$} &
%   \textbf{0.842${\scriptscriptstyle\pm 0.038}$} &
%   \textbf{32.5${\scriptscriptstyle\pm 1.6}$} &
%   \textbf{0.015${\scriptscriptstyle\pm 0.004}$} &
%   \textbf{0.752${\scriptscriptstyle\pm 0.053}$} &
%   \textbf{29.3${\scriptscriptstyle\pm 1.6}$} &
%   \textbf{0.030${\scriptscriptstyle\pm 0.007}$} \\ \hline
% SSL ALL Oversamp. &
%   \multicolumn{1}{r}{0.955${\scriptscriptstyle\pm 0.015}$} &
%   \multicolumn{1}{r}{38.5${\scriptscriptstyle\pm 2.5}$} &
%   \multicolumn{1}{r}{0.004${\scriptscriptstyle\pm 0.002}$} &
%   \multicolumn{1}{r}{0.891${\scriptscriptstyle\pm 0.030}$} &
%   \multicolumn{1}{r}{34.6${\scriptscriptstyle\pm 2.0}$} &
%   \multicolumn{1}{r}{0.009${\scriptscriptstyle\pm 0.004}$} &
%   \multicolumn{1}{r}{0.807${\scriptscriptstyle\pm 0.039}$} &
%   \multicolumn{1}{r}{31.3${\scriptscriptstyle\pm 1.6}$} &
%   \multicolumn{1}{r}{0.019${\scriptscriptstyle\pm 0.006}$} &
%   \multicolumn{1}{r}{0.712${\scriptscriptstyle\pm 0.051}$} &
%   \multicolumn{1}{r}{28.1${\scriptscriptstyle\pm 1.6}$} &
%   \multicolumn{1}{r}{0.040${\scriptscriptstyle\pm 0.009}$} \\
% JSSL Oversamp. &
%   \textbf{0.968${\scriptscriptstyle\pm 0.012}$} &
%   \textbf{41.0${\scriptscriptstyle\pm 2.1}$} &
%   \textbf{0.002${\scriptscriptstyle\pm 0.001}$} &
%   \textbf{0.919${\scriptscriptstyle\pm 0.026}$} &
%   \textbf{36.7${\scriptscriptstyle\pm 1.9}$} &
%   \textbf{0.006${\scriptscriptstyle\pm 0.002}$} &
%   \textbf{0.842${\scriptscriptstyle\pm 0.038}$} &
%   \textbf{32.6${\scriptscriptstyle\pm 1.6}$} &
%   \textbf{0.014${\scriptscriptstyle\pm 0.004}$} &
%   \textbf{0.749${\scriptscriptstyle\pm 0.052}$} &
%   \textbf{29.2${\scriptscriptstyle\pm 1.6}$} &
%   \textbf{0.031${\scriptscriptstyle\pm 0.008}$} \\ \hline
% SSL (q=0.5) &
%   0.957${\scriptscriptstyle\pm 0.014}$ &
%   39.1${\scriptscriptstyle\pm 2.4}$ &
%   0.003${\scriptscriptstyle\pm 0.002}$ &
%   0.895${\scriptscriptstyle\pm 0.027}$ &
%   35.0${\scriptscriptstyle\pm 1.9}$ &
%   0.008${\scriptscriptstyle\pm 0.003}$ &
%   0.817${\scriptscriptstyle\pm 0.038}$ &
%   31.7${\scriptscriptstyle\pm 1.6}$ &
%   0.017${\scriptscriptstyle\pm 0.005}$ &
%   0.733${\scriptscriptstyle\pm 0.050}$ &
%   28.9${\scriptscriptstyle\pm 1.6}$ &
%   0.033${\scriptscriptstyle\pm 0.008}$ \\
% SSL ALL (q=0.5) &
%   0.956${\scriptscriptstyle\pm 0.015}$ &
%   39.0${\scriptscriptstyle\pm 2.5}$ &
%   0.004${\scriptscriptstyle\pm 0.002}$ &
%   0.897${\scriptscriptstyle\pm 0.029}$ &
%   35.2${\scriptscriptstyle\pm 2.0}$ &
%   0.008${\scriptscriptstyle\pm 0.003}$ &
%   0.816${\scriptscriptstyle\pm 0.039}$ &
%   31.7${\scriptscriptstyle\pm 1.6}$ &
%   0.018${\scriptscriptstyle\pm 0.005}$ &
%   0.715${\scriptscriptstyle\pm 0.052}$ &
%   28.3${\scriptscriptstyle\pm 1.6}$ &
%   0.038${\scriptscriptstyle\pm 0.009}$ \\
% JSSL (q=0.5) &
%   \textbf{0.965${\scriptscriptstyle\pm 0.015}$} &
%   \textbf{39.7${\scriptscriptstyle\pm 2.8}$} &
%   \textbf{0.003${\scriptscriptstyle\pm 0.002}$} &
%   \multicolumn{1}{r}{\textbf{0.919${\scriptscriptstyle\pm 0.026}$}} &
%   \multicolumn{1}{r}{\textbf{36.6${\scriptscriptstyle\pm 1.8}$}} &
%   \multicolumn{1}{r}{\textbf{0.006${\scriptscriptstyle\pm 0.002}$}} &
%   \multicolumn{1}{r}{\textbf{0.842${\scriptscriptstyle\pm 0.037}$}} &
%   \multicolumn{1}{r}{\textbf{32.6${\scriptscriptstyle\pm 1.6}$}} &
%   \multicolumn{1}{r}{\textbf{0.014${\scriptscriptstyle\pm 0.004}$}} &
%   \multicolumn{1}{r}{\textbf{0.742${\scriptscriptstyle\pm 0.052}$}} &
%   \multicolumn{1}{r}{\textbf{29.0${\scriptscriptstyle\pm 1.6}$}} &
%   \multicolumn{1}{r}{\textbf{0.032${\scriptscriptstyle\pm 0.007}$}} \\ \hline
% SSL (w=10) &
%   0.954${\scriptscriptstyle\pm 0.015}$ &
%   38.4${\scriptscriptstyle\pm 2.5}$ &
%   0.004${\scriptscriptstyle\pm 0.002}$ &
%   0.893${\scriptscriptstyle\pm 0.029}$ &
%   34.7${\scriptscriptstyle\pm 2.0}$ &
%   0.009${\scriptscriptstyle\pm 0.004}$ &
%   0.815${\scriptscriptstyle\pm 0.038}$ &
%   31.6${\scriptscriptstyle\pm 1.6}$ &
%   0.018${\scriptscriptstyle\pm 0.006}$ &
%   0.726${\scriptscriptstyle\pm 0.048}$ &
%   28.5${\scriptscriptstyle\pm 1.6}$ &
%   0.036${\scriptscriptstyle\pm 0.008}$ \\
% SSL ALL (w=10) &
%   0.954${\scriptscriptstyle\pm 0.015}$ &
%   38.4${\scriptscriptstyle\pm 2.5}$ &
%   0.004${\scriptscriptstyle\pm 0.002}$ &
%   0.890${\scriptscriptstyle\pm 0.029}$ &
%   34.6${\scriptscriptstyle\pm 2.0}$ &
%   0.009${\scriptscriptstyle\pm 0.004}$ &
%   0.805${\scriptscriptstyle\pm 0.039}$ &
%   31.2${\scriptscriptstyle\pm 1.6}$ &
%   0.020${\scriptscriptstyle\pm 0.006}$ &
%   0.710${\scriptscriptstyle\pm 0.052}$ &
%   28.1${\scriptscriptstyle\pm 1.6}$ &
%   0.040${\scriptscriptstyle\pm 0.009}$ \\
% JSSL (w=10) &
%   \textbf{0.958${\scriptscriptstyle\pm 0.016}$} &
%   \textbf{38.7${\scriptscriptstyle\pm 2.6}$} &
%   \textbf{0.004${\scriptscriptstyle\pm 0.002}$} &
%   \textbf{0.916${\scriptscriptstyle\pm 0.026}$} &
%   \textbf{36.4${\scriptscriptstyle\pm 1.8}$} &
%   \textbf{0.006${\scriptscriptstyle\pm 0.002}$} &
%   \textbf{0.839${\scriptscriptstyle\pm 0.038}$} &
%   \textbf{32.5${\scriptscriptstyle\pm 1.6}$} &
%   \textbf{0.015${\scriptscriptstyle\pm 0.004}$} &
%   \textbf{0.748${\scriptscriptstyle\pm 0.052}$} &
%   \textbf{29.2${\scriptscriptstyle\pm 1.6}$} &
%   \textbf{0.031${\scriptscriptstyle\pm 0.008}$} \\ \hline
% \end{tabular}%
% }
% }
% \label{tab:ablation}
% \end{table*}
% % ------------------------------TABLE-------------------------------------------%

% Summarized in \Table{ablation}, we calculated the average evaluation metrics on the test set for our alternative configurations experiments, providing additional context to the JSSL approach.  These experiments consistently showcased the superior performance of JSSL over SSL setups, in line with our prior observations.  Interestingly, variations in the training hyper-parameters for JSSL, such as oversampling, partitioning ratio ($q=0.5$), and ACS window size ($w=10$), did not yield significant improvements or deteriorations in performance, except for an observable improvement in average pSNR at $R=2$.

% Regarding the SSL setups, an observable enhancement was witnessed for 8$\times$ and 16$\times$ accelerated data when adopting a fixed partitioning ratio $q=0.5$ or a larger ACS window of $w^2=10^2$ pixels. However, this improvement was particularly evident in the SSL setup using solely the (subsampled) proxy dataset. Furthermore, the inclusion of proxy datasets within SSL configurations (SSL ALL) did not yield improvements in reconstruction performance, consistent with our earlier findings in the comparative study.

% For further assessment, we provide in Appendix \ref{sec:ap6} box plots illustrating comprehensively the performance metrics, as well as sample reconstructions for each setup considered in the alternative configurations study.

