How to prove the convexity of
$$f (W) = \operatorname{Tr}(WLW^T)$$
where $L$ is a (positive semidefinite) Laplacian matrix?
I need to prove this to build a convex function for optimization in a machine learning method.
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That is just a number, numbers are not convex. What function are you talking about $W \mapsto $, $L \mapsto $, $(W,L) \mapsto$? – copper.hat Sep 25 '19 at 04:07
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Thanks copper, the function is about $f(W)=\operatorname{Tr}(WLW^T)$ – 胡丹青 Sep 25 '19 at 05:39
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Since $f(W) = \sum_k e_k^T W L W^T e_k$, it suffices to show that $g(W) = v^T W L W^T v $ is convex for any $v$.
Since $L \ge 0$, we have $2 x^Ty \le x^TLx+y^TLy $ for any $x,y$.
Then \begin{eqnarray} g(t W_1+(1-t)W_2) &=& t^2g(W_1)+(1-t)^2g(W_2)+2t(1-t)vW_1LW_2^T v \\ &\le& t^2g(W_1)+(1-t)^2g(W_2)+t(1-t)(g(W_1)+g(W_2)) \\ &=& t g(W_1)+(1-t)g(W_2) \end{eqnarray} and so $g$, and hence $f$, is convex.
A simpler way is to note that since $L \ge 0$ we can write $L=C^TC$ and so $g(v) =\|CW^Tv\|^2$. Since $W \mapsto CW^T v$ is linear, $\|\cdot\|$ is convex and so $W \mapsto \|CW^Tv\|$ is convex. Since $t \mapsto t^2$ is convex and nondecreasing on $t \ge 0$ we see that $g$ is convex.
An even simpler way is to let $\phi(t) = g(W+tH)$ and compute $\phi''(0) = 2 v^THLH^Tv $ which is $\ge 0$, since $L \ge 0$. Then $t \mapsto g(t W_1+(1-t)W_2)$ is convex and hence so is $g$.
copper.hat
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