论文公式.md

$$L_{z}=\sum ^{V}{v=1}L{z}^{v}=\sum ^{V}_{v=1}\left| X^{m}-D^{m}\left( E^{m}\left( X^{m}\right) \right) \right| _{2}^{2}$$

$$S^{F}=\dfrac{<Q,\widetilde{Q}^{T} >}{\left| Q\right| _{2}\left| \widetilde Q^{T}\right| _{2}}$$

$$P^{F}=F_{cla}(Q)$$

$$\mathbf{W}{ij}^{F} = \begin{cases} 1 & \text{if } i = j, \ p{i}^{F} \cdot p_{j}^{F} & \text{if } i \neq j \text{ and } p_{i}^{F} \cdot p_{j}^{F} \geq \tau, \ 0 & \text{otherwise,} \end{cases}$$

$$\mathcal{L}{F} = - \mathbf{W}{ii}^{F} \log \left( \frac{\exp(S_{ii}^{F})}{\sum \exp(S_{ij}^{F})} \right)\sum_{j=1, j \neq i}^{N} -\mathbf{W}{ij}^{F} \log \left( \frac{\exp(S{ij}^{F})}{\sum \exp(S_{ij}^{F})} \right).$$

$$\mathcal{L}{FAC} = \frac{1}{2} | S^{F} - I |{2}^{2}$$

$$P^{k}=F_{cla}(Z^{k})$$

$$\mathbf{W}{ij}^{kk^{'}} = \begin{cases} 1 & \text{if } i = j, \ p{i}^{k} \cdot p_{j}^{k^{'}} & \text{if } i \neq j \text{ and } p_{i}^{k} \cdot p_{j}^{k^{'}} \geq \tau, \ 0 & \text{otherwise,} \end{cases}$$

$$S^{kk^{'}}=\dfrac{<Z^{k},(Z^{k^{'}})^{T} >}{\left| Z^{k}\right| _{2}\left| (Z^{k^{'}})^{T}\right| _{2}}$$

$$\mathcal{L}{causal} = \sum{\substack{k, k' \in S \ k < k'}} \left( -\mathbf{W}{ii}^{kk^{'}} \log \left( \frac{\exp(S{ii}^{kk^{'}})}{\sum \exp(S_{ij}^{kk^{'}})} \right)- \sum_{j=1, j \neq i}^{N} \mathbf{W}{ij}^{kk^{'}} \log \left( \frac{\exp(S{ij}^{kk^{'}})}{\sum \exp(S_{ij}^{kk^{'}})} \right) \right),$$

$$\mathcal{L}{align} = \frac{1}{\binom{2}{\lvert S \rvert}} \sum{\substack{k, k' \in S \ k < k'}} | \mathbf{W}^{F} - \mathbf{W}^{kk^{'}} |_{2}^{2},$$

$$\mathcal{L} = \mathcal{L}{z} + \alpha \mathcal{L}{F} + \beta \mathcal{L}{causal} + \mu \mathcal{L}{align},$$

$$U=\sum ^{V}{v=1}\dfrac{e^{w^{v}{\alpha}}}{\sum ^{V}{k=1}e^{w^{k}{\alpha}}}H^{v }$$

$$\beta(H^{v}) = \mu(H^{v}) + \epsilon_\mu \Sigma_\mu(H^{v}), \quad \epsilon_\mu \sim \mathcal{N}(0, 1),$$

$$\gamma(H^{v}) = \sigma(H^{v}) + \epsilon_\sigma \Sigma_\sigma(H^{v}), \quad \epsilon_\sigma \sim \mathcal{N}(0, 1),$$​

$$\widetilde H^{v} = \gamma(H^{v}) \left( \frac{H^{v} - \mu(H^{v})}{\sigma(H^{v})} \right) + \beta(H^{v}).$$