# Quadratic Multilinear Discriminant Analysis for Tensorial Data Classification
## 5 Discussion  
Our proposed method depends on the choice of group action, the regularization parameter \(\epsilon\), and the maximum number of iterations. In all the experiments we performed, our method seems to be mostly affected by the choice of group action (as we would expect). This might suggest a cross-validation approach to choose the best group action for a given dataset. On the other end, on the synthetic data--perhaps due to their symmetry--\(\text{SL}_{n_{1}}\times\text{SL}_{n_{2}}\times\text{SL}_{n_{3}}\) consistently achieved the best results. In the case of ECG tensors from the PTB dataset, Table 9 reports average AUCs for each possible group action: \(\text{SL}_{6}\times\text{SL}_{5}\times\text{SL}_{12}\) is still best or close to being best. Therefore, it seems that in practice choosing \(\text{SL}_{n}\) action in each mode is always a safe choice, and results in a less costly model. On the other hand, it is possible that for tensors whose dimensions vary greatly from one mode to another, and for which each mode has a significantly different nature, another group action would perform much better. In that case, we would still be limited to eight choices (for third-order tensors). In any case, varying the regularization parameter \(\epsilon\) or the maximum number of iterations typically results in negligible variations in performance. Tables 10 and 11 report variations in AUC for the ECG tensors from the PTB dataset. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Dimensions** & **[1,1,1]** & **[2,2,2]** & **[3,3,3]** & **[4,4,4]** & **[5,5,5]** \\ \hline AUC & 0.82 & 0.81 & 0.82 & 0.82 & 0.85 \\ \hline SD & 0.05 & 0.08 & 0.06 & 0.10 & 0.09 \\ \hline
**Dimensions** & **[6,5,4]** & **[6,5,5]** & **[6,5,6]** & **[6,5,7]** & **[6,5,8]** \\ \hline AUC & 0.85 & 0.84 & 0.71 & 0.52 & 0.57 \\ \hline SD & 0.09 & 0.07 & 0.12 & 0.16 & 0.10 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Performance (mean AUC and standard deviation) of ManTDA on tensors from the PTB dataset as the dimensions \([I,J,K]\) increase (10 iterations). \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Method** & **Clean** & **SNR = -10** & **SNR = -30** \\ \hline KNMDA & **0.94 (0.03)** & **0.92 (0.03)** & **0.88 (0.04)** \\ DATEReg & 0.90 (0.04) & 0.81 (0.06) & 0.80 (0.06) \\ DATER & 0.90 (0.04) & 0.83 (0.05) & 0.82 (0.04) \\ CMDA & 0.91 (0.03) & 0.80 (0.08) & 0.79 (0.07) \\ SVM & 0.91 (0.03) & 0.85 (0.06) & 0.85 (0.05) \\ LDA & 0.68 (0.07) & 0.59 (0.07) & 0.56 (0.07) \\ ManTDA & 0.87 (0.05) & 0.72 (0.10) & 0.70 (0.12) \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Classification of tensors extracted from ECG recordings, with different signal-to-noise ratios (mean AUC and standard deviation). The bold values represent the best performance for each column. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Method** & **Clean** & **SNR = -10** & **SNR = -30** \\ \hline STS & **0.95 (0.03)** & 0.91 (0.03) & 0.86 (0.04) \\ TST & 0.88 (0.06) & 0.86 (0.05) & 0.86 (0.05) \\ TSS & 0.93 (0.03) & **0.92 (0.03)** & **0.89 (0.03)** \\ SST & 0.91 (0.05) & 0.87 (0.05) & 0.87 (0.05) \\ TTS & 0.94 (0.03) & 0.90 (0.03) & 0.88 (0.03) \\ SSS & 0.94 (0.03) & **0.92 (0.03)** & 0.88 (0.04) \\ TTT & 0.89 (0.06) & 0.88 (0.06) & 0.87 (0.05) \\ STT & 0.91 (0.04) & 0.87 (0.06) & 0.84 (0.06) \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Average AUCs (and standard deviations) for all possible choices of group actions (S for \(\text{SL}_{n}\) and T for \(\text{T}_{n}\)) on PTB tensors. For example, STS stands for \(\text{SL}_{6}\times\text{T}_{5}\times\text{SL}_{12}\). The bold values represent the best performance for each column. We note that other MDA methods would require us to test, in principle, \(n_{1}n_{2}n_{3}\) dimensions; significantly more than the 8 possible combinations of groups for KNMDA (and similarly for higher order tensors). Additionally, another known issue of correctly choosing the reduced dimensions is that the performance is not monotonic as dimensionality increases (as shown in [11]).