## 3 Materials and Methods
### Regional Topic Model and Its Online Inference Algorithm
After Markov chain converged, the arithmetic average value of sample sufficient statistics \(N_{j\omega rt}\) is the estimation of \(\mathbb{E}_{\mathbf{\eta}_{j}}\big{[}N_{j\omega rt}\big{]}\)  
\[\mathbb{E}_{\mathbf{\eta}_{j}}\big{[}N_{j\omega rt}\big{]}=\mathbb{E}_{\mathbf{ \eta}_{j}}\Big{[}\sum_{n=1}^{N_{j}}\mathbb{I}\big{(}\mathbf{x}_{jn}=w\big{)} \mathbb{I}\big{(}\mathbf{r}_{jn}=r\big{)}\mathbb{I}\big{(}\mathbf{t}_{jn}=t \big{)}\Big{]}\approx\ \frac{1}{S}\sum_{s=1}^{S}N_{j\omega rt}^{(s)} \tag{20}\]  
2. Global Stochastic Variational Inference  
In the \(t\)th stochastic iteration, the state space of Markov chain is constructed by the sample sufficient statistics \(N_{(B_{t})\omega rt}=\sum_{j\in B_{t}}N_{j\omega rt}\) of a stochastic small batch documents \(B_{t}\), the contribution of \(B_{t}\) to natural gradient of global variational parameters is  
\[\widetilde{N}_{(B_{t})\omega rt}=\mathbb{E}_{\mathbf{\eta}_{(B_{t})}}\Big{[}N_{( B_{t})\omega rt}\Big{]}\approx\frac{1}{S}\sum_{s=1}^{S}N_{(B_{t})\omega rt}^{(s)}= \frac{1}{S}\sum_{s=1}^{S}\!\!\sum_{j\in B_{t}}\!\!N_{j\omega rt}^{(s)} \tag{21}\]At this point, the update amount \(\xi_{ir}^{(t)}\), \(\phi_{r}^{(t)}\) and \(\hat{\mu}_{\omega rt}^{(t)}\) of local small batch documents \(B_{t}\) respect to global variational parameters is  
\[\left\{ \begin{array}{ll}\xi_{ir}^{(t)}&\propto\exp\biggl{\{}\mathbb{E}_{\
u_{ r}^{(t-1)}}[\log\rho_{r}]+(J/|B_{t}|)\sum_{u=1}^{\Omega}\sum_{t=1}^{T}\widehat{N}_{(B_{t} )\omega rt}^{(t)}\mathbb{E}_{\mu_{\omega rt}^{(t-1)}}[\log\theta_{\omega rt}] \biggr{\}}\\ \
u_{r}^{(t)}&=\beta~{}+~{}\sum_{i=1}^{I}\xi_{ir}^{(t)}\\ \hat{\mu}_{\omega rt}^{(t)}&=\lambda~{}+~{}(J/|B_{t}|)\mathbb{E}_{\sigma^{(t)} }\bigl{[}\widehat{N}_{(B_{t})\omega rt}^{(t)}\bigr{]}\end{array} \right. \tag{22}\]  
Then, the variational expectation of \(\widehat{N}_{j\omega rt}\) or \(\widehat{N}_{(B_{t})\omega rt}\) obtained from local MCMC inference respect to global variational parameters is that  
\[ \begin{array}{ll}\mathbb{E}_{\sigma}\left[\mathbb{E}_{\eta_{j}}\bigl{[}N_{ j\omega rt}\bigr{]}\right]&=\mathbb{E}_{\sigma}\left[\widehat{N}_{j\omega rt }\right]=\mathbb{E}_{\sigma}\Bigl{[}\sum_{n=1}^{N_{j}}\mathbb{I}\bigl{(} \mathbf{x}_{jn}=\omega\bigr{)}\mathbb{I}\bigl{(}\mathbf{\tilde{r}}_{jn}=r \bigr{)}\mathbb{I}\bigl{(}\mathbf{\tilde{t}}_{jn}=t\bigr{)}\,\right]\\ &=\sum_{n=1}^{N_{j}}\sum_{i=1}^{I}\sum_{r=1}^{R}\xi_{ir}\mathbb{I}\bigl{(} \mathbf{x}_{jn}=\omega\bigr{)}\mathbb{I}\bigl{(}\mathbf{i}_{jn}=i\bigr{)} \mathbb{I}\bigl{(}\mathbf{\tilde{r}}_{i}=r\bigr{)}\mathbb{I}\bigl{(}\mathbf{ \tilde{t}}_{jn}=t\bigr{)}\\ &=\sum_{n=1}^{N_{j}}\xi_{ir}\mathbb{I}\bigl{(}\mathbf{x}_{jn}=\omega\bigr{)} \mathbb{I}\bigl{(}\mathbf{\tilde{t}}_{jn}=t\bigr{)}\end{array}  \tag{23}\]  
When the number of motion topics \(T\), motion regions \(R\) or words \(\Omega\) is large, \(N_{j}\) observed samples of document \(j\) is allocated to a large \(T\times R\times\Omega\) dimensions hybrid parameters matrix \(\theta=\{\theta_{\omega rt}\}_{\omega=1,r=1,t=1}^{\Omega,R,T}\), which make sufficient statistics \(N_{j\omega rt}\) of many samples be zero. Then, \(\mathbb{E}_{\eta_{j}}\bigl{[}N_{j\omega rt}\bigr{]}\) estimated by MCMC sampling is a sparse matrix. Therefore, the amount of computations is decreased and computing speed is improved because of the sparsity. Given the above description, the specific description of HSVG algorithm is shown in Algorithm 1.