# Global-Entropy Driven Exploration with Distributed Models under Sparsity Constraints
## 3 Distributed Cooperative Exploration
### Centralized Exploration
Define now:  
\[\mathbf{A}\triangleq\lambda\mathbf{\Psi}_{\mathcal{A}}^{T}\mathbf{\Psi}_{\mathcal{A}}+ \widehat{\mathbf{\Delta}}_{\mathcal{A}},\quad\mathbf{c}(\mathbf{\tilde{x}})^{T}\triangleq \lambda\mathbf{\Psi}_{\mathcal{A}}^{T}\mathbf{\varphi}(\mathbf{X}_{M},\mathbf{\tilde{x}}), \quad b(\mathbf{\tilde{x}})\triangleq\lambda\mathbf{\varphi}(\mathbf{X}_{M},\mathbf{\tilde{x}}) ^{T}\mathbf{\varphi}(\mathbf{X}_{M},\mathbf{\tilde{x}}). \tag{23}\]  
With the definitions in (23), the cost function (22) can be re-cast as:  
\[\mathbf{\hat{x}}=\underset{\mathbf{x}\in\mathcal{X},\mathbf{\tilde{x}}\in\mathcal{X}_{ \mathcal{I}}}{\arg\max}\log\left| \begin{bmatrix}\mathbf{A}&\mathbf{c}(\mathbf{\tilde{x}}) \\ \mathbf{c}(\mathbf{\tilde{x}})^{T}&b(\mathbf{\tilde{x}})\end{bmatrix} +\lambda \begin{bmatrix} \mathbf{\psi}_{\mathcal{A}}(\mathbf{x})\\ \varphi(\mathbf{x},\mathbf{\tilde{x}})\end{bmatrix}\begin{bmatrix}\mathbf{\psi}_{\mathcal{A }}(\mathbf{x})^{T}&\varphi(\mathbf{x},\mathbf{\tilde{x}})\end{bmatrix} \right|, \tag{24}\]  
where we use \(\mathbf{\psi}_{\mathcal{A}}(\mathbf{x})=\mathbf{\psi}_{\mathcal{A}}(\mathbf{x},\mathbf{X}_{N})\) to lighten up the notation. We make use of the matrix determinant lemma to bring (24) in the following form:\[\widehat{x}=\underset{\mathbf{x}\in\mathcal{X},\mathbf{\xi}\in\mathcal{X}_{ \mathcal{I}}}{\arg\max}\log\left| \begin{bmatrix}\mathbf{A}&\mathbf{c}(\tilde{\mathbf{x}})\\ \mathbf{c}(\tilde{\mathbf{x}})^{T}&b(\tilde{\mathbf{x}})\end{bmatrix} \right|\] \[\qquad\qquad\qquad+\log\left|1+\lambda \begin{bmatrix}\mathbf{\psi}_{ \mathcal{A}}(\mathbf{x})\\ \varphi(\mathbf{x},\tilde{\mathbf{x}})\end{bmatrix} ^{T} \begin{bmatrix}\mathbf{A}&\mathbf{c}( \tilde{\mathbf{x}})\\ \mathbf{c}(\tilde{\mathbf{x}})^{T}&b(\tilde{\mathbf{x}})\end{bmatrix} ^{-1} \begin{bmatrix} \mathbf{\psi}_{\mathcal{A}}(\mathbf{x})\\ \varphi(\mathbf{x},\tilde{\mathbf{x}})\end{bmatrix} \right|. \tag{25}\]  
The first log-term in (25) can be shown to be equal to \(\log\left| \begin{bmatrix}\mathbf{A}&\mathbf{c}(\tilde{\mathbf{x}})\\ \mathbf{c}(\tilde{\mathbf{x}})^{T}&b(\tilde{\mathbf{x}})\end{bmatrix} \right|=\log|\mathbf{A}|+ \log|q(\tilde{\mathbf{x}})|\), where \(q(\tilde{\mathbf{x}})=b(\tilde{\mathbf{x}})-\mathbf{c}(\tilde{\mathbf{x}})^{T}\mathbf{A}^{-1}\mathbf{ c}(\tilde{\mathbf{x}})\) is the Schur complement of \(b(\tilde{\mathbf{x}})\) in the corresponding matrix under the logarithm. To simplify the second log-term in (25), we note that:  
\[ \begin{bmatrix}\mathbf{A}&\mathbf{c}(\tilde{\mathbf{x}})\\ \mathbf{c}(\tilde{\mathbf{x}})^{T}&b(\tilde{\mathbf{x}})\end{bmatrix} ^{-1}= \begin{bmatrix} \mathbf{A}^{-1}-\mathbf{A}^{-1}\mathbf{c}(\tilde{\mathbf{x}})q(\tilde{\mathbf{x}})^{-1}\mathbf{c}( \tilde{\mathbf{x}})^{T}\mathbf{A}^{-1}&-\mathbf{A}^{-1}\mathbf{c}(\tilde{\mathbf{x}})/q(\tilde\mathbf{{x}})\\ -\mathbf{c}(\tilde{\mathbf{x}})^{T}\mathbf{A}^{-1}/q(\tilde{\mathbf{x}})&1/q(\tilde{\mathbf{x}}) \end{bmatrix} . \tag{26}\]  
Then,  
\[\log\left|1+\lambda \begin{bmatrix}\mathbf{\psi}_{\mathcal{A}}(\mathbf{x}) \\ \varphi(\mathbf{x},\tilde{\mathbf{x}})\end{bmatrix} ^{T} \begin{bmatrix}\mathbf{A}&\mathbf{c}( \tilde{\mathbf{x}})\\ \mathbf{c}(\tilde{\mathbf{x}})^{T}&b(\tilde{\mathbf{x}})\end{bmatrix} ^{-1} \begin{bmatrix} \mathbf{\psi}_{\mathcal{A}}(\mathbf{x})\\ \varphi(\mathbf{x},\tilde{\mathbf{x}})\end{bmatrix}