In the scene of nonnegative matrix factorization, $f(X_1, X_2) = \| Y - X_1 X_2 \|_F$ is not convex, but both $f(X_1)$ given fixed $X_2$ and $f(X_2)$ given fixed $X_1$ are convex, enabling us to perform Expectation-Maximization style alternating optimization.
In case that $X_2 = X_1^T$, we have a new objective $f(X) = \| Y - XX^T\|_F$. Is this function convex with respect to $X$? ($X$ is not necessarily a square matrix but $X \geq 0$ (entrywise), whereas $Y$ is a square and symmetric matrix such that $Y \geq 0$) Could you explain why?
Thanks in advance for the help.
It's not convex in the scalar case (consider $x\mapsto |1-x^2|$ for instance), so it's in general not convex in higher dimensional cases --- at least not when $Y$ and $X$ are scalar multiples of the identity matrix.