# Quadratic Multilinear Discriminant Analysis for Tensorial Data Classification
## 4 Experimental Results
### Synthetic Data
#### 4.1.2. HOSVD Based Data  
In this experiment, we generate three-way tensors by creating the building blocks of the HOSVD structure similarly to the way in which matrices are created in [3] for testing: in fact, we follow the same construction but adapted their code to create third-order tensors. We generate two distinct cores \(\mathcal{G}_{1}\) and \(\mathcal{G}_{2}\) of size \(3\times 3\times 3\) with standard normal random entries for two classes, as well as matrices \(U_{1},U_{2},U_{3},V_{1},V_{2},V_{3}\) of size \(10\times 3\) with orthogonal columns. We then generate observations for each class by adding noise to these cores, according to the following structure:  
\[\mathcal{X}_{i}=U_{1}\times_{1}U_{2}\times_{2}U_{3}\times_{3}(\sigma\mathcal{ G}_{1}+\eta\mathcal{N}_{i})+25(V_{1}\times_{1}V_{2}\times_{2}V_{3}\times_{3} \mathcal{E}_{i}),\]  
\[\mathcal{Y}_{i}=U_{1}\times_{1}U_{2}\times_{2}U_{3}\times_{3}(\sigma\mathcal{ G}_{2}+\eta\mathcal{N}_{i})+25(V_{1}\times_{1}V_{2}\times_{2}V_{3}\times_{3} \mathcal{E}_{i})\]  
where \(\mathcal{N}_{i}\) and \(\mathcal{E}_{i}\) are tensors of size \(3\times 3\times 3\) with standard normal random entries. The higher the value of \(\sigma\) and the lower the value of \(\eta\), the easier it is to discriminate between classes. Table 3 reports AUCs for three decreasing levels of discriminability (the same three levels as in [3]); for each level of noise, we computed averages over 10 iterations using 40 tensors for training and 100 for testing.  
For KNMDA, the SL-action was used in all modes, with imposed regularization with parameter \(\epsilon=1\) and a maximum of 10 iterations. For DATER, DATEReig, CMDA, ManTDA, and ManPDA were projected onto the space \(\mathbb{R}^{3\times 3\times 3}\), which is the true dimension of the underlying cores and therefore gives the best results.  
\begin{table}
\begin{tabular}{c c c c} \hline
**Method** & \(\eta=1,\rho=1\) & \(\eta=2,\rho=1\) & \(\eta=3,\rho=1\) \\ \hline KNMDA & **1.00 (0.00)** & **0.75 (0.05)** & **0.60 (0.04)** \\ DATEReig & 0.92 (0.04) & 0.52 (0.06) & 0.52 (0.04) \\ DATER & 0.89 (0.07) & 0.51 (0.03) & 0.52 (0.05) \\ CMDA & 0.93 (0.04) & 0.53 (0.06) & 0.51 (0.05) \\ SVM & **1.00 (0.00)** & 0.61 (0.04) & 0.53 (0.05) \\ LDA & 0.95 (0.02) & 0.57 (0.03) & 0.52 (0.05) \\ \hline \end{tabular}
\end{table}
Table 1: CP data with \(r_{1}=3,r_{2}=3\), size \(10\times 10\times 10\). Different columns report AUC (and standard deviation) for different levels of noise. The bold values represent the best performance for each column.  
\begin{table}
\begin{tabular}{c c c c} \hline
**Method** & \(\eta=1,\rho=3\) & \(\eta=1,\rho=5\) & \(\eta=1,\rho=7\) \\ \hline KNMDA & **0.92 (0.02)** & **0.82 (0.03)** & **0.73 (0.05)** \\ DATEReig & 0.68 (0.07) & 0.62 (0.06) & 0.59 (0.05) \\ DATER & 0.65 (0.07) & 0.61 (0.07) & 0.59 (0.06) \\ CMDA & 0.70 (0.05) & 0.62 (0.05) & 0.60 (0.06) \\ SVM & 0.80 (0.04) & 0.70 (0.05) & 0.66 (0.05) \\ LDA & 0.69 (0.04) & 0.62 (0.04) & 0.60 (0.05) \\ \hline \end{tabular}
\end{table}
Table 2: CP data with \(r_{1}=3\), \(r_{2}=3\), size \(5\times 5\times 5\). Different columns report AUC (and standard deviation) for different levels of noise. The bold values represent the best performance for each column.