# Global-Entropy Driven Exploration with Distributed Models under Sparsity Constraints
## 3 Distributed Cooperative Exploration
### Centralized Exploration
For the active columns, we will denote this row vector as \(\mathbf{\psi}_{\mathcal{A}}(\mathbf{x},\mathbf{X}_{N})=[\varphi(\mathbf{x},\mathbf{x}_{1}),\dots, \varphi(\mathbf{x},\mathbf{x}_{N})]_{\mathcal{A}}^{T}\), where subscript \(\mathcal{A}\) indicates the restriction on \(N\) columns to the active subset. Additionally, a new measurement can potentially \"activate\" a feature, i.e., it might lead to some element in \(\mathbf{w}\) becoming non-zero at the next estimation stage of the algorithm (it is reasonable to assume that at most one feature is activated per measurement as the number of degrees of freedom should not grow faster than the number of measurements). An activated feature will result in an extension of \(\mathbf{\
abla}_{\mathcal{A}}\) with a new column vector \(\mathbf{\varphi}(\mathbf{X}_{M},\mathbf{\tilde{x}})=[\varphi(\mathbf{X}_{1},\mathbf{\tilde{x}}), \dots,\varphi(\mathbf{X}_{M},\mathbf{\tilde{x}})]^{T}\), with \(\mathbf{\tilde{x}}\in\mathcal{X}_{\mathcal{I}}\), where \(\mathcal{X}_{\mathcal{I}}\) denotes the set of possible inactive feature locations. Finally, due to the measurement at a location \(\mathbf{x}\in\mathcal{X}\) a new design matrix \(\widetilde{\mathbf{\Psi}}_{\mathcal{A}}(\mathbf{x},\mathbf{\tilde{x}})\) can be constructed as follows:  
\[\widetilde{\mathbf{\Psi}}_{\mathcal{A}}(\mathbf{x},\mathbf{\tilde{x}})= \begin{bmatrix} \mathbf{\
abla}_{\mathcal{A}}&\mathbf{\varphi}(\mathbf{X}_{M},\mathbf{\tilde{x}})\\ \mathbf{\psi}_{\mathcal{A}}(\mathbf{x},\mathbf{X}_{N})^{T}&\varphi(\mathbf{x},\mathbf{\tilde{x}}) \end{bmatrix} , \tag{20}\]  
and the corresponding \"new\" covariance is then given as \(\widetilde{\mathbf{C}}=\left(\lambda\widetilde{\mathbf{\Psi}}_{\mathcal{A}}(\mathbf{ \tilde{x}},\mathbf{\tilde{x}})^{T}\widetilde{\mathbf{\Psi}}_{\mathcal{A}}(\mathbf{\tilde{ x}},\mathbf{\tilde{x}})+\text{diag}\{\widehat{\mathbf{\Delta}}_{\mathcal{A}},0\} \right)^{-1}\). Here, setting the regularization for the new feature weight to zero puts no regularization on a potentially new placed feature. If the regularization is too high, we might place features at already measured positions, cf. Section 4.1. The D-optimality criterion yields a measurement location \(\mathbf{\tilde{x}}\) that minimizes \(\log\det\widetilde{\mathbf{C}}\), i.e.,  
\[\mathbf{\tilde{x}}=\underset{\mathbf{x}\in\mathcal{X},\mathbf{\tilde{x}}\in\mathcal{X}_{ \mathcal{I}}}{\arg\min}\log\left|\lambda\widetilde{\mathbf{\Psi}}_{\mathcal{A}}( \mathbf{x},\mathbf{\tilde{x}})^{T}\widetilde{\mathbf{\Psi}}_{\mathcal{A}}(\mathbf{x},\mathbf{ \tilde{x}})+ \begin{bmatrix}\widehat{\mathbf{\Delta}}_{\mathcal{A}}&\\ &0\end{bmatrix} \right|^{-1}. \tag{21}\]  
We then proceed by inserting (20) into (21) and simplifying the result, which leads to:  
\[\mathbf{\tilde{x}}=\underset{\mathbf{x}\in\mathcal{X},\mathbf{\tilde{x}}\in \mathcal{X}_{\mathcal{I}}}{\arg\max}\log\left| \begin{bmatrix}\lambda\mathbf{\Psi} _{\mathcal{A}}^{T}\mathbf{\Psi}_{\mathcal{A}}+\widehat{\mathbf{\Delta}}_{\mathcal{A} }&\lambda\mathbf{\Psi}_{\mathcal{A}}^{T}\mathbf{\varphi}(\mathbf{X}_{M},\mathbf{\tilde{x}})\\ \lambda\mathbf{\varphi}(\mathbf{X}_{M},\mathbf{\tilde{x}})^{T}\mathbf{\Psi}_{\mathcal{A}}& \lambda\mathbf{\varphi}(\mathbf{X}_{M},\mathbf{\tilde{x}})^{T}\mathbf{\varphi}(\mathbf{X}_{M}, \mathbf{\tilde{x}})\end{bmatrix} \right.\] \[\left.+\lambda \begin{bmatrix}\mathbf{\psi}_{\mathcal{A}}(\mathbf{x},\mathbf{X }_{N})\\ \varphi(\mathbf{x},\mathbf{\tilde{x}})\end{bmatrix}\begin{bmatrix}\mathbf{\psi}_{\mathcal{A }}(\mathbf{x},\mathbf{X}_{N})\\ \varphi(\mathbf{x},\mathbf{\tilde{x}})\end{bmatrix} ^{T}\right|. \tag{22}\]  
As we see, (22) must be evaluated for each potential measurement position \(\mathbf{x}\) and for each potential feature position \(\mathbf{\tilde{x}}\). This requires computing a determinant of a large matrix for every test position and a possible active feature, a computationally costly numerical problem. This computation can be implemented more efficiently by exploiting the structure of (22) as follows.