# Robust Bilinear Probabilistic Principal Component Analysis
## 3 Robust BPPCA  
### The Model  
In this section, we develop a robust model by replacing the matrix variate Gaussian distribution in BPPCA with the matrix variate \(t\) distribution with degrees of freedom \(\
u\) defined in (9) to deal with 2-D data sets. Specifically, the proposed robust bilinear probabilistic principal analysis model (RBPPCA for short) is defined as  
\[ \begin{cases}X=CZR^{\text{T}}+W+C\mathcal{E}_{r}+\mathcal{E}_{c}R^{\text{T}}+ \mathcal{E},\\ \mu\sim Ga(v/2,\
u/2),\quad\mu_{c}\mu_{r}=\mu^{-1},\\ Z|\mu\sim\mathcal{N}_{d_{c},d_{r}}(0_{q_{c}\times q_{r}},\mu_{c}I_{c}\mu_{r},\mu_{r}I_{q_{r}}),\ \mathcal{E}_{c}|\mu\sim\mathcal{N}_{d_{c},d_{r}}(0_{d_{c}\times q_{r}},\mu_{c} \sigma_{c}^{2}I_{d_{c}},\mu_{r}I_{q_{r}}),\\ \mathcal{E}_{r}|\mu\sim\mathcal{N}_{q_{c},d_{r}}(0_{q_{c}\times d_{r}},\mu_{c} I_{q_{r}},\mu_{r}\sigma_{c}^{2}I_{d_{r}}),\ \mathcal{E}|\mu\sim\mathcal{N}_{d_{c},d_{r}}(0_{d_{c}\times d_{r}},\mu_{c} \sigma_{c}^{2}I_{d_{c}},\mu_{r}\sigma_{c}^{2}I_{d_{r}}).\end{cases}  \tag{11}\]  
As BPPCA [12], in the RBPPCA model (11), \(\mathcal{E}_{c}\in\mathbb{R}^{d_{c}\times q_{r}}\), \(\mathcal{E}_{r}\in\mathbb{R}^{d_{c}\times d_{r}}\) and \(\mathcal{E}\in\mathbb{R}^{d_{c}\times d_{r}}\) are the column, row and common noise matrices, respectively, \(Z\in\mathbb{R}^{q_{c}\times q_{r}}\) is the latent matrix, and these are assumed to be independent of each other, and the mean matrix, and the column and row factor loading matrices are \(W\in\mathbb{R}^{d_{c}\times d_{r}}\), \(C\in\mathbb{R}^{d_{c}\times q_{c}}\) and \(R\in\mathbb{R}^{d_{r}\times q_{r}}\), respectively. Similarly to BPPCA [12], the parameters \(C\), \(R\), \(\sigma_{c}\) and \(\sigma_{r}\) can not be uniquely identified, but the interested subspaces spanned by the columns of \(C\) and \(R\) are unique. The reader is referred to ([12], Appendix B) for details. The difference from BPPCA is that the noise matrices \(\mathcal{E}_{c}\), \(\mathcal{E}_{r}\) and \(\mathcal{E}\) and latent matrix variate \(Z\) in the RBPPCA model (11) are supposed matrix variate \(t\) distributions by (10), i.e.,  
\[\mathcal{E}_{c}\sim\mathcal{T}_{d_{c},d_{r}}(\
u,0_{d_{c}\times d _{r}},\sigma_{c}^{2}I_{d_{c}},q_{r}),\quad\mathcal{E}_{r}\sim\mathcal{T}_{q_{c },d_{r}}(\
u,0_{q_{c}\times d_{r}},I_{q_{c}},\sigma_{r}^{2}I_{d_{r}}),\] \[\mathcal{E}\sim\mathcal{T}_{d_{c},d_{r}}(\
u,0_{d_{c}\times d_{r} },\sigma_{c}^{2}I_{d_{c}},\sigma_{r}^{2}I_{d_{r}}),\;\;Z\sim\mathcal{T}_{q_{c },q_{r}}(\
u,0_{q_{c}\times q_{r}},I_{q_{c}},I_{q_{r}}).\]  
It follows by (11) that  
\[CZR^{\mathrm{T}}|\mu\sim\mathcal{N}_{d_{c},d_{r}}(0_{d_{c}\times d _{r}},\mu_{c}C\mathrm{C}^{\mathrm{T}},\mu_{r}RR^{\mathrm{T}}),\] \[C\mathcal{E}_{r}|\mu\sim\mathcal{N}_{d_{c},d_{r}}(0_{d_{c}\times d _{r}},\mu_{c}C\mathrm{C}^{\mathrm{T}},\mu_{r}\sigma_{r}^{2}I_{d_{r}}),\] \[\mathcal{E}_{c}R^{\mathrm{T}}|\mu\sim\mathcal{N}_{d_{c},d_{r}}(0_ {d_{c}\times d_{r}},\mu_{c}\sigma_{c}^{2}I_{d_{c}},\mu_{r}RR^{\mathrm{T}}).\]  
Consequently, \(X|\mu\sim\mathcal{N}_{d_{c},d_{r}}(W,\mu_{c}\Sigma_{c},\mu_{r}\Sigma_{r})\) where  
\[\Sigma_{c}=C\mathrm{C}^{\mathrm{T}}+\sigma_{c}^{2}I_{d_{c}}\quad\text{and} \quad\Sigma_{r}=RR^{\mathrm{T}}+\sigma_{r}^{2}I_{d_{r}}. \tag{12}\]  
That means the random matrix \(X\) follows the matrix variate \(t\) distribution, i.e., \(X\sim\mathcal{T}_{d_{c},d_{r}}(\
u,W,\Sigma_{c},\Sigma_{r})\). In addition, as shown in [23], the conditional distribution \(\mu|X\) which is also required in our later estimation of model parameters is a Gamma distribution, i.e.,  
\[\mu|X\sim Ga\bigg{(}\frac{\
u+d_{c}d_{r}}{2},\frac{\
u+\rho}{2}\bigg{)}, \tag{13}\]  
where \(\rho=\text{tr}\left(\Sigma_{c}^{-1}(X-W)\Sigma_{r}^{-1}(X-W)^{\mathrm{T}}\right)\). By introducing two latent matrix variates \(Y^{\prime}\in\mathbb{R}^{q_{c}\times d_{r}}\) and \(Y^{\prime}_{\epsilon}\in\mathbb{R}^{d_{c}\times d_{r}}\), the RBPPCA model (11) can be rewritten as  
\[