Given a binary square matrix $A \in \{0, 1\}^{n \times n}$ and $1 \leq k \leq n$, we aim to compute the error of outer-product-based low-rank approximation. Formally, we aim to compute $$ \min_{X \in \mathbb{R}^{n \times k}} \lVert \sigma(XX^T) - A \rVert, $$ where $\sigma$ is a normalization function making sure that each entry of $XX^T$ is between $0$ and $1$.
In the simplest case, $\sigma$ can be simply an entry-wise clipping function $\sigma(x) = \max(\min(x, 1), 0)$. Or even simpler, results without $\sigma$ are also appreciated. In more complicated cases, $\sigma$ can be nonlinear, e.g., entry-wise sigmoid.
Can we have results at least for the simplest case? Any matrix norm is acceptable to me.