## 3 Preliminaries for the Proposed Method
### Bilateral-Projection Based 2 DPCA  
In [49], a bilateral-projection based 2-D principal component analysis (B2 DPCA) has been proposed, which minimizes the following energy functional:  
\[[\mathbf{P}_{opt},\mathbf{Q}_{opt}]=\operatorname*{argmin}_{\mathbf{P}, \mathbf{Q}}\|\mathbf{X}-\mathbf{PCQ}^{T}\|_{\mathrm{F}}^{2}, \tag{1}\]  
where \(\mathbf{X}\in R^{n\times m}\) is the two dimensional image, \(\mathbf{P}\in R^{m\times l}\) and \(\mathbf{Q}\in R^{n\times r}\) are the left- and right- multiplying projection matrices, respectively, and \(\mathbf{C}=\mathbf{P}^{T}\mathbf{X}\mathbf{Q}\) is the extracted feature matrix for the image \(\mathbf{X}\). The optimal projection matrices \(\mathbf{P}_{opt}\) and \(\mathbf{Q}_{opt}\) are simultaneously constructed, where \(\mathbf{P}_{opt}\) projects the column vectors of \(\mathbf{X}\) to a subspace, while \(\mathbf{Q}_{opt}\) projects the row vectors of \(\mathbf{X}\) to another one. It has been shown in [49], that the advantage of the bilateral projection over the unilateral-projection scheme is that \(\mathbf{X}\) can be represented effectively with smaller number of coefficients than in the unilateral case, i.e., a small-sized matrix \(\mathbf{C}\) can well represent the image \(\mathbf{X}\). This means that the bilateral-projection effectively removes the redundancies among both rows and columns of the image. Furthermore, since  
\[\mathbf{C}=\mathbf{P}^{T}\mathbf{X}\mathbf{Q}=\left[ \begin{array}{ cccc}\mathbf{p}_{\mathrm{I}}^{T}\mathbf{X}\mathbf{q}_{1}&\mathbf{p}_{\mathrm{I}}^{T} \mathbf{X}\mathbf{q}_{2}&...&\mathbf{p}_{\mathrm{I}}^{T}\mathbf{X}\mathbf{q}_ {r}\\ \mathbf{p}_{\mathrm{I}}^{T}\mathbf{X}\mathbf{q}_{1}&\mathbf{p}_{\mathrm{I}}^{T} \mathbf{X}\mathbf{q}_{2}&...&\mathbf{p}_{\mathrm{I}}^{T}\mathbf{X}\mathbf{q}_ {r}\\ \vdots&\vdots&\vdots&\vdots\\ \mathbf{p}_{\mathrm{I}}^{T}\mathbf{X}\mathbf{q}_{1}&\mathbf{p}_{\mathrm{I}}^{T} \mathbf{X}\mathbf{q}_{2}&...&\mathbf{p}_{\mathrm{I}}^{T}\mathbf{X}\mathbf{q}_ {r}\\ \vdots&\vdots&\vdots&\vdots\\ \mathbf{<X},\mathbf{p}_{\mathrm{I}}\mathbf{q}_{1}^{T}>&\mathbf{<X},\mathbf{p}_{ \mathrm{I}}\mathbf{q}_{2}^{T}>&...&\mathbf{<X},\mathbf{p}_{\mathrm{I}} \mathbf{q}_{r}^{T}>\end{array} \right] \tag{2}\]  
it can be seen that the components of \(\mathbf{C}\) are the 2-D projections of the image \(\mathbf{X}\) onto the 2-D planes \(\mathbf{p}_{1}\mathbf{q}_{1}^{T}\), \(\mathbf{p}_{1}\mathbf{q}_{2}^{T}\),...\(\mathbf{p}_{\mathrm{I}}\mathbf{q}_{r}^{T}\) made up by the outer products of the column vectors of \(\mathbf{P}\) and \(\mathbf{Q}\). The 2-D planes have a rank of one, since they are the outer products of two 1-D vectors. Therefore, the fact that \(\mathbf{X}\) can be well represented by a small-sized \(\mathbf{C}\) also implies the fact that \(\mathbf{X}\) can be well represented by a few rank-1 2-D planes, i.e., only a few 1-Dvectors \(\mathbf{p}_{1},...\mathbf{p}_{l},\mathbf{q}_{1},...\mathbf{q}_{r}\), where \(l<<m\) and \(r<<n\). In the case of (1), the learned 2-D planes try to minimize the loss function  
\[L=\|\mathbf{X}-\mathbf{PCQ}^{T}\|_{F}^{2}, \tag{3}\]  
i.e., try to learn to best approximate \(\mathbf{X}\). A natural question arises, if good rank-1 2-D planes can be obtained to minimize other loss functions too, e.g., loss functions related to the image classification problem, such as  
\[L=\|y_{true}-y(\mathbf{X},\mathbf{P},\mathbf{C},\mathbf{Q})\|_{F}^{2}, \tag{4}\]  
where \(y_{true}\) denotes the true classification label for a certain input image \(\mathbf{X}\), and \(y(\mathbf{X},\mathbf{P},\mathbf{C},\mathbf{Q})\) is the output of the network constituted by the outer products of the column vectors in the learned matrices \(\mathbf{P}\) and \(\mathbf{Q}\). In this paper, we extend this case to the rank-1 3-D filter case, where the rank-1 3-D filter is constituted as the outer product of three column vectors from three different learned matrices, and show that good parameters can be learned for the image classification task.