Assume I have $X \in \{-1, 1\}^{m\times n}$. Lets denote the columns of $X$ as the $m$ dimensional integer vectors $x_1,...,x_n \in \{-1,1\}^m$.
Given $Y \in \{-1, 1\}^{n\times n}$ where $Y_{ij} \in \{-m, -m+1,..., m-1, m\}$. How can one approach to solve the 'fitting-like' problem:
\begin{equation} \min_X f(Y - X^TX) \end{equation} where $f$ is a element-wise convex function of $Y - X^TX$, e.g., a frobenius norm. To put it differently, I would like to find $x_i$ and $x_j$ such that $\sum_{i,j} f(Y_{ij} - x_i^Tx_j)$ is minimized.
What methods, approaches should one take? Notice that $Y_{ij}$ and $x_i^Tx_j$ have the same co-domain.