# Quadratic Multilinear Discriminant Analysis for Tensorial Data Classification
## 5 Discussion
This costly step of finding best dimensions to project onto is not necessary in KNMDA, since it looks for an optimal change of coordinates instead of an optimal projection. On the other hand, one potential downside of the fact that KNMDA does not project the data onto a smaller dimensional space is that the most costly operations involve the computation of inverse matrices of dimensions \(n_{1}\), \(n_{2}\), and \(n_{3}\), which could be computationally heavier. However, in our experiments on relatively small tensors it is significantly faster than any other tensor method we tested. This is shown in Table 12, reporting average times in seconds for training and testing several methods over HOSVD-based data; training times are averages over 40 samples and testing times are averages over 100 samples. One limitation of KNMDA is that it might not be suited for tensors of very large dimension: in this scenario, a standard tensor dimensionality reduction preprocessing step would be useful. Furthermore, most MDA methods require random initialization, which means they can provide different results for each run on the same dataset, and incur the additional cost of trying out different initializations for each run. KNMDA's results on the contrary are completely determined by the input and do not change with different runs of the algorithm. This fact and the previous discussion make a case for the stability of the proposed method. Based on the experiments on synthetic data, we can see that generating data with both fundamental tensorial decompositions (CP and HOSVD), KNMDA outperforms the other methods we compared it with. In cases where KNMDA is comparable to one or more other methods, it will still often outperform it as the amount of noise is increased. Another impressive feature of KNMDA is that its performance is consistent over tensors of very different natures. For example, linear SVM performs better than all other MDA methods over CP-based data, but it is close to random guessing on HOSVD-based data. This latter type of data is where MDA methods seem strongest, with manifold-based techniques \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Maximum Iterations** & **1** & **3** & **5** & **10** & **50** & **100** \\ \hline AUC & 0.90 & 0.93 & 0.93 & 0.93 & 0.93 & 0.93 \\ \hline SD & 0.04 & 0.03 & 0.03 & 0.03 & 0.03 & 0.03 \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Performance of the proposed method (mean AUC and standard deviation) on ECG tensors as the maximum number of iterations varies. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & **KNMDA** & **DATER** & **DATEReig** & **CMDA** & **ManTDA** \\ \hline \multirow{2}{*}{train} & 0.0603 & 0.6252 & 1.3007 & 1.8484 & 30.3623 \\  & (0.0207) & (0.0997) & (0.1953) & (0.1836) & (5.5336) \\ \hline \multirow{2}{*}{test} & 0.1043 & 0.4238 & 0.6347 & 0.1836 & 0.1209 \\  & (0.0146) & (0.2660) & (0.1963) & (0.0446) & (0.0350) \\ \hline \hline \end{tabular}
\end{table} 
Table 12: Average execution times in seconds (and standard deviations) for HOSVD tensors, on a training set of 40 tensors and a test set of 100 tensors. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Value of Parameter \(\epsilon\)** & **0.1** & **0.5** & **1** & **2** & **5** & **10** \\ \hline AUC & 0.93 & 0.93 & 0.93 & 0.93 & 0.92 & 0.92 \\ \hline SD & 0.04 & 0.04 & 0.04 & 0.04 & 0.04 & 0.04 \\ \hline \hline \end{tabular}
\end{table} 
Table 11: Performance of the proposed method (mean AUC and standard deviation) on ECG tensors as the regularization parameter \(\epsilon\) varies. performing better than the other MDA methods, concordant with the results from [3]. However, KNMDA performs better than both tensor-based methods and linear SVM on both of these datasets. Finally, in the case of sparsity patterns, KNMDA achieves a perfect classification already at a level of noise at which other methods are unable to distinguish between classes.