# Functional Connectome Fingerprinting Through Tucker Tensor Decomposition
## 2 Materials and Methods
### Tucker Decomposition of Functional Comectomes
\[\mathbf{A}^{(n)}\in\mathbb{R}^{I_{a}\times R_{N}}\text{ and column-wise orthonormal}\]  
where the factor matrices \(\mathbf{A}^{(1)}\times_{2}\mathbf{A}^{(2)}\cdots\times_{N}\mathbf{A}^{(N)}\) are commonly initialized via the ST-HOSVD of \(\mathbf{\mathcal{X}}\). By iteratively optimizing each factor matrix while keeping the others fixed, HOOI provides a better fit as measured by the norm of the difference between the true and estimated tensors compared to ST-HOSVD, but for a higher computational cost. However, since HOOI is not guaranteed to converge to a global optimum nor to a stationary point [43; 44] and does not provide substantial fingerprinting improvements in comparison to ST-HOSVD (see Table 2), ST-HOSVD is the chosen algorithm to compute the Tucker decomposition of FCs. When applied to tensors that exhibit partial (or full) symmetries, HOSVD preserves the symmetric structure of the tensor [41]. Hence, for a tensor consisting of one session (e.g., test sessions) of participants' FC matrices, the Tucker decomposition of \(\mathbf{\mathcal{X}}\in\mathbb{R}^{M\times M\times N}\) can be reformulated as:  
\[\min_{\mathbf{\mathcal{G}},\mathbf{B},\mathbf{P}} ||\mathbf{\mathcal{X}}-\mathbf{\mathcal{G}}\times_{1}B\times_{2}B\times _{3}\mathbf{P}||\] s.t. \[\mathbf{\mathcal{G}}\in\mathbb{R}^{R_{1}\times R_{1}\times R_{2}}\] \[B\in\mathbb{R}^{M\times R_{1}}\text{ and column-wise orthonormal}\] \[\mathbf{P}\in\mathbb{R}^{N\times R_{2}}\text{ and column-wise orthonormal}\]  
where the factor matrices \(\mathbf{B}\) and \(\mathbf{P}\) obtained via HOSVD contain, respectively, brain parcellation and participant-specific information and the ranks \(R_{1}\) and \(R_{2}\) express the compression levels of brain parcellation and participant-specific information. While solving the optimization problem presented in (2), the brain parcellation ranks were chosen with a step size of \(\frac{(Parcellation-14)}{4}\). In addition to the previous parcellation ranks, we also performed a full-rank decomposition. Thus, for a parcellation granularity of 414, for example, the brain parcellation ranks were set to \(R_{1}=\{100,200,300,400,414\}\). In contrast, for all parcellation granularities explored, the participant ranks were set to \(R_{2}=\{50,100,150,200,250,300,350,400,426\}\). Under the hypothesis that the functional connectivity patterns of a participant are, to some extent, reproducible across scanning sessions, we fix the core tensor \(\mathbf{\mathcal{G}}\) and brain \begin{table}
\begin{tabular}{l c c c c c} \hline \hline
**fMRI Conditions** & **FCs** & **PCA** & **CP** & **ST-HOSVD** & **HOOI** \\ \hline Emotion (166 TPs) & 0.57 & 0.63 & 0.64 & **0.90** & **0.90** \\ Relational (222 TPs) & 0.79 & 0.84 & 0.90 & **0.99** & **0.99** \\ Gambling (243 TPs) & 0.84 & 0.86 & 0.93 & **0.98** & **0.98** \\ Social (264 TPs) & 0.88 & 0.88 & 0.92 & **0.97** & **0.97** \\ Motor (274 TPs) & 0.82 & 0.82 & 0.75 & **0.96** & **0.96** \\ Language (306 TPs) & 0.91 & 0.90 & 0.92 & **0.98** & **0.98** \\ Working Memory (395 TPs) & 0.86 & 0.84 & 0.92 & **0.97** & **0.97** \\ Resting-state (1190 TPs) & 0.92 & 0.97 & 0.99 & **1.00** & **1.00** \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Matching rate comparison between (baseline) FCs and distinct dimensionality reduction methods for parcellation granularity of 414. Bold values highlight the highest matching rates among all techniques for each fMRI condition, which are presented in ascending order of scanning length (“TPs” represent the number of time points in the time series of each condition). parcellation factor matrix \(B\) derived from the Tucker decomposition of tensor \(\mathbf{\mathcal{X}}\) and estimate the participant factor matrix \(\mathbf{Q}\) of the tensor \(\mathbf{\mathcal{Y}}\) comprising FCs from another data acquisition session (e.g., retest session). By doing so, we aim to detect a consistent presence of underlying cohort-level functional connectivity patterns across different data acquisition sessions for each participant.