# Quadratic Multilinear Discriminant Analysis for Tensorial Data Classification
## 4 Experimental Results
### Synthetic Data
#### 4.1.3 Classification of Sparsity Patterns  
Inspired by [20], we considered another kind of data. For fixed factor matrices \(A\), \(B\) and \(C\), we say that the congruence class determined by them is the set of all tensors of the form \(\mathcal{X}=\sum_{i=1}^{r}\sigma_{i}a_{i}\circ b_{i}\circ c_{i}\) for any \(\sigma_{1},\ldots,\sigma_{r}\in\mathbb{R}\). It can be shown that congruence sets arising from distinct factor matrices do not intersect ([20], Remark 1). We can therefore consider the natural classification problem that consists of discriminating between different congruence classes. Let \(e_{j}\) be the \(j\)-th canonical basis vector of \(\mathbb{R}^{l}\). Let \(\mathcal{L}_{j}\) be \(e_{j}\circ e_{j}\circ e_{j}\). We generate tensors according to two classes: \(\mathcal{X}_{i}=x_{i}\mathcal{L}_{1}+y_{i}\mathcal{L}_{2}+z_{i}\mathcal{L}_{3 }+E_{i}\) and \(\mathcal{Y}_{j}=x_{j}\mathcal{L}_{4}+y_{j}\mathcal{L}_{5}+z_{j}\mathcal{L}_{6 }+E_{j}\), where \(x_{i}\), \(y_{i}\) and \(z_{i}\) are i.i.d. from a zero-mean Gaussian distribution with variance \(1-\beta^{2}\), and the entries of the noise tensors \(E_{i}\) are i.i.d. from a zero-mean Gaussian distribution with variance \(\beta^{2}\). Table 4 reports the average AUC over 20 runs for several methods, where 40 samples in each class were used for training and 100 in each class for testing. In KNMDA, SL-action was used in all modes, with imposed regularization with parameter \(\epsilon=1\) and a maximum of 10 iterations. For DATER, DATEReig and CMDA we projected onto the space \(\mathbb{R}^{3\times 3\times 3}\) (other choices of tensor spaces of equal size in each mode performed equivalently). ManTDA similarly performed at the level of a random guess. It is worth noting that our method seems especially suitable for this type of classification problem, even with very few training samples. In [20], this experiment is used to test a kernel designed for tensorial data, which improves the classification results compared to kernels designed for vectorial data. The authors reported an AUC of 0.86 (0.04) with only \(M=10\) training samples, whereas our method already achieves perfect classification with \(M=6\) training samples, as Table 5 shows. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Method** & \(\beta^{2}=0.05\) & \(\beta^{2}=0.15\) & \(\beta^{2}=0.25\) \\ \hline KNMDA & **1.00 (0.00)** & **0.99 (0.01)** & **0.76 (0.05)** \\ DATEReig & 0.51 (0.06) & 0.48 (0.04) & 0.49 (0.06) \\ DATER & 0.51 (0.06) & 0.49 (0.06) & 0.52 (0.04) \\ CMDA & 0.50 (0.06) & 0.50 (0.05) & 0.51 (0.05) \\ SVM & 0.51 (0.06) & 0.50 (0.06) & 0.50 (0.06) \\ LDA & 0.50 (0.06) & 0.52 (0.06) & 0.49 (0.06) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Classification of Sparsity Patterns. The bold values represent the best performance for each column. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Method** & \(\sigma=1\), \(\eta=1\) & \(\sigma=0.5\), \(\eta=\sqrt{3}\) & \(\sigma=0.25\), \(\eta=\sqrt{3}\) \\ \hline KNMDA & **1.00 (0.00)** & **0.87 (0.03)** & **0.58 (0.06)** \\ DATEReig & 0.78 (0.10) & 0.62 (0.04) & 0.55 (0.08) \\ DATER & 0.87 (0.14) & 0.64 (0.08) & 0.53 (0.06) \\ CMDA & 0.99 (0.02) & 0.73 (0.07) & 0.55 (0.06) \\ SVM & 0.58 (0.13) & 0.52 (0.03) & 0.48 (0.05) \\ LDA & 0.89 (0.05) & 0.62 (0.07) & 0.53 (0.04) \\ ManTDA & 0.90 (0.06) & 0.62 (0.07) & 0.56 (0.06) \\ ManPDA & **1.00** (0.01) & 0.72 (0.05) & 0.56 (0.06) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: HOSVD data. Different columns report AUC (and standard deviation) for decreasing levels of discriminability of the classes. The bold values represent the best performance for each column.