We have a deterministic symmetric binary matrix (e.g., adjacency matrix) $A \in \{0, 1\}^{n \times n}$. Given a vector $a \in [0, 1]^{n}$, we compute its outer product $\tilde{A} = a a^T \in [0, 1]^{n \times n}$, where $\tilde{A}_{ij} = a_i a_j$. We see $\tilde{A}$ as a probability matrix and we aim to maximize the likelihood that $\tilde{A}$ takes the value of $A$, i.e., we aim to find $$ \arg\max_{a \in [0, 1]^{n}} \prod_{i,j} \Pr[\operatorname{Bernoulli}(a_ia_j) = A_{ij}], $$ where $\Pr[\operatorname{Bernoulli}(a_ia_j) = A_{ij}] = \tilde{A}_{ij} A_{ij} + (1 - \tilde{A}_{ij}) (1 - A_{ij})$.
Can we have a closed-form solution for the above optimization problem?