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Consider the functions $f:\mathbb{R}^n\to\mathbb{R}$, $g:\mathbb{R}^m\to\mathbb{R}$ and the non-square matrix $A \in \mathbb{R}^{m\times n}$ with $m>n$.

Let $x\in\mathbb{R}^n$ and consider the change of variables $y = Ax$. Let $f(x) = g(Ax)$.

If $g(y)$ is strictly convex in $y$, under what conditions on $A$ can we say that $f(x)$ is strictly convex in $x$?

My attempt:

The Hessian of $g$ is positive definite, that is $H_g\succ0$. Is the Hessian of $f$ equal to $A^TH_gA$?

Would $A$ being full rank ensure that $H_g\succ0\Rightarrow A^TH_gA\succ0$?

user140268
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  • As your question is written there is no relationship between $f$ and $g$. Perhaps you want to assume $f(x) = g(Ax)$? – Paul Siegel Apr 04 '14 at 07:33
  • Yes, thank you, edited. – user140268 Apr 04 '14 at 07:33
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    Then yes, the right condition is that $A$ has full rank. $f$ is necessarily convex, and the only way $f$ can degenerate is for $A$ to have kernel. Your computations with Hessians look OK, but you can also just write down inequalities if you want to allow $g$ to be non-differentiable. – Paul Siegel Apr 04 '14 at 07:39

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