Table 1:

Notations in ${\mathcal{L}}_{\text{ELBO}-\text{VAE}}\left(x\right)$.

Notations in the ELBO

θ: parameters in the decoder. ψ: parameters in the encoder. N: the total number of observations. T: the total number of clusters. D: the dimension of the latent representation z. Σ : diag(σ2(x; ψ)). xij: the jth dimension of the nth observation. yn: cluster membership for the nth observation. $p\left({\mathit{y}}_n=k\right)={\gamma }_{ik}$, $N_k={\sum }_{n=1}^N{\gamma }_{nk}$. L: the number of Monte Carlo samples in Stochastic Gradient Variational Bayes (SGVB). ${\widehat{z}}_n=\frac{1}{L}{\sum }_{l=1}^L{\mathit{z}}_n^{\left(l\right)}\cdot {\overline{z}}_k=\frac{1}{N_k}{\sum }_{n=1}^N{\gamma }_{nk}{\widehat{z}}_n$. ${\mathit{U}}_k=\frac{1}{{\mathcal{N}}_k}{\sum }_{n=1}^N{\gamma }_{nk}\left({\widehat{z}}_n-{\overline{z}}_k\right){\left({\widehat{z}}_n-{\overline{z}}_k\right)}^T$. βk = β0 + Nk: the posterior scalar precision in NW distribution. ${\mathit{m}}_k=\frac{1}{{\beta }_k}\left({\beta }_0{\mathit{m}}_0+N_k{\overline{z}}_k\right)$: the posterior mean of cluster k. $W_k^{-1}=W_0^{-1}+N_k{S}_k+\frac{{\beta }_0{N}_k}{{\beta }_0+N_k}\left({\overline{z}}_k-{\mathit{m}}_0\right){\left({\overline{z}}_k-{\mathit{m}}_0\right)}^T$. ${\nu }_k={\nu }_0+N_k$: the kth posterior degrees of freedom of NW.