# Robust Bilinear Probabilistic Principal Component Analysis
## 2 Preliminaries  
Let \(X\in\mathbb{R}^{d_{c}\times d_{r}}\), \(M\in\mathbb{R}^{d_{c}\times d_{r}}\), \(\Omega_{c}\in\mathbb{R}^{d_{c}\times d_{c}}\) and \(\Omega_{r}\in\mathbb{R}^{d_{r}\times d_{r}}\), and the probability density function of the random variable \(\mu\) having a Gamma distribution with parameters \(\alpha\) and \(\beta\), i.e., \(\mu\sim Ga(\alpha,\beta)\), be  
\[p(\mu)=\beta^{\alpha}\mu^{\alpha-1}\exp(-\beta\mu)/\Gamma(\alpha), \tag{5}\]  
where \(\Gamma(\cdot)\) is the Gamma function, i.e.,  
\[\Gamma(\alpha)=\int_{0}^{+\infty}\tau^{\alpha-1}\exp(-\tau)\,d\tau. \tag{6}\]  
Analogously to the process of tPPCA in [14], we derive the matrix variate \(t\) distribution in this paper by considering  
\[\text{vec}(X)\sim\int_{0}^{+\infty}\mathcal{N}\bigg{(}\text{vec}(M),\frac{ \Omega_{r}\otimes\Omega_{c}}{\mu}\bigg{)}p(\mu)\,d\mu,\]  
where \(\mu\sim Ga(\
u/2,\
u/2)\). Let \(\delta=\text{tr}\left(\Omega_{c}^{-1}(X-M)\Omega_{r}^{-1}(X-M)^{\text{T}}\right)\). We have  
\[\int_{0}^{+\infty}\mathcal{N}\bigg{(}\text{vec}(M),\frac{\Omega _{r}\otimes\Omega_{c}}{\mu}\bigg{)}p(\mu)d\mu\] \[= \int_{0}^{+\infty}(2\pi)^{-\frac{d_{c}d_{r}}{2}}\bigg{|}\frac{ \Omega_{r}\otimes\Omega_{c}}{\mu}\bigg{|}^{-\frac{1}{2}}\exp\biggl{(}-\frac{ \delta\mu}{2}\biggr{)}\Big{(}\frac{\
u}{2}\Big{)}^{\frac{\
u}{2}}\mu^{\frac{ \
u}{2}-1}\exp\Bigl{(}-\frac{\
u}{2}\mu\Bigr{)}\frac{1}{\Gamma\big{(}\frac{ \
u}{2}\big{)}}\,d\mu\] \[= \frac{(2\pi)^{-\frac{d_{c}d_{r}}{2}}|\Omega_{c}|^{-\frac{d_{r}}{2 }}|\Omega_{r}|^{-\frac{d_{r}}{2}}\left(\frac{\
u}{2}\right)^{\frac{\
u}{2}}}{ \Gamma\big{(}\frac{\
u}{2}\big{)}}\int_{0}^{+\infty}\mu^{\frac{\
u+d_{c}d_{r }}{2}-1}\exp\biggl{(}-\frac{(\delta+\
u)\mu}{2}\biggr{)}\,d\mu. \tag{7}\]Let \(\tau=\frac{(\delta+v)\mu}{2}\). Then, \(\mu=\frac{2\tau}{v+\delta}\) and \(d\mu=\frac{2}{v+\delta}\,d\tau\). Therefore, (7) can be rewritten as  
\[\int_{0}^{+\infty}\mathcal{N}\bigg{(}\text{vec}(M),\frac{\Omega_{ r}\otimes\Omega_{c}}{\mu}\bigg{)}p(\mu)d\mu\] \[= \frac{(2\pi)^{-\frac{d\phi_{L}}{2}}|\Omega_{c}|^{-\frac{d\phi_{L} }{2}}|\Omega_{r}|^{-\frac{d\phi_{L}}{2}}\big{(}\frac{v}{2}\big{)}^{\frac{v}{2}}} {\Gamma\big{(}\frac{v}{2}\big{)}}\bigg{(}\frac{2}{v+\delta}\bigg{)}^{\frac{v+ \delta\phi_{L}}{2}}\int_{0}^{+\infty}\tau^{\frac{v+\delta\phi_{L}}{2}-1}\exp(- \tau)\,d\tau\] \[= \frac{(2\pi)^{-\frac{d\phi_{L}}{2}}|\Omega_{c}|^{-\frac{d\phi_{L}} {2}}|\Omega_{r}|^{-\frac{d\phi_{L}}{2}}\big{(}\frac{v}{2}\big{)}^{\frac{v}{2}} }{\Gamma\big{(}\frac{v}{2}\big{)}}\bigg{(}\frac{2}{v+\delta}\bigg{)}^{\frac{v+ \delta\phi_{L}}{2}}\,\Gamma\bigg{(}\frac{v+d\phi_{L}}{2}\bigg{)}\] \[= \frac{\pi^{-\frac{d\phi_{L}}{2}}|\Omega_{c}|^{-\frac{d\phi_{L}}{2 }}|\Omega_{r}|^{-\frac{d\phi_{L}}{2}}\tau^{\frac{v}{2}}\Gamma\big{(}\frac{v+ \delta\phi_{L}}{2}\big{)}}{\Gamma\big{(}\frac{v}{2}\big{)}}v^{-\frac{v+\delta \phi_{L}}{2}}\bigg{(}1+\frac{\delta}{v}\bigg{)}^{-\frac{v+d\phi_{L}}{2}}\] \[\Gamma(\alpha)=\int_{0}^{+\infty}\tau^{\alpha-1}\exp(-\tau)\,d\tau. \tag{6}\]0  
In this paper, if the pdf of the random matrix \(X\) is  
\[\Gamma(\alpha)=\int_{0}^{+\infty}\tau^{\alpha-1}\exp(-\tau)\,d\tau. \tag{6}\]1  
then the random matrix \(X\) is said to follow the matrix variate \(t\) distribution with degrees of freedom \(\
u\), and is denoted by  
\[\Gamma(\alpha)=\int_{0}^{+\infty}\tau^{\alpha-1}\exp(-\tau)\,d\tau. \tag{6}\]2  
In particular, if \(d_{c}=1\) or \(d_{r}=1\), then the matrix variate \(t\) distribution degenerates to the multivariate \(t\) distribution. As the classical multivariate \(t\) distribution, another favorite perspective on the matrix variate \(t\) distribution which is critical to our later developments, is to treat \(\mu\) as a latent variable, then the conditional distribution of \(X|\mu\) is a matrix variate Gaussian distribution by (8), i.e.,  
\[\Gamma(\alpha)=\int_{0}^{+\infty}\tau^{\alpha-1}\exp(-\tau)\,d\tau. \tag{6}\]3  
where \(\mu_{c}\mu_{r}=\mu^{-1}\).