Convexity of the matrix function (Schur Product Operator norm)

Question

I have a function defined as

$$ \mathbf{f(X)} = \min_{~~~~U,V\\\mathbf{X = UV'}} \max_i \mathbf{\|U_i\|} \max_i \mathbf{\|V_i\|} \\ \text{where } \mathbf{U_i} ~\text{and}~ \mathbf{V_i} ~ \text{are row vectors and } \|\cdot\| ~\text{is the}~ \ell_2 ~\text{norm} $$ $\mathbf{X}$ is a real rectangular matrix of dimension $m \times n$.

How can I prove that the function is convex ?

According to the paper , this quantity is the optimum of the Schur product operator norm of a matrix: $f(A) = \max_{X :\|X\|=1} \|A \circ X\|$.

Answer 1

Using the equivalent description that you mentioned, it is easier to show that $f$ is convex.

For a fixed $X$, the function $$ g_X(A) = \| A \circ X \| $$ is convex, because $A\mapsto A\circ X$ is linear and norms are convex.

Then, $$ f(A) = \sup_{X:\|X\|=1} g_X(A) $$ means that $f$ is the supremum of convex functions and therefore convex itself.