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Assume $w, a_1,a_2 \in \mathbb{R}^d$ and $\sigma = \dfrac{1}{1+e^{-x}}$ the sigmoid function. Is the following squared difference a convex function?

$$J(w)= (\sigma(w^Ta_1)\times \sigma(w^T a_2)-c)^2$$

where $c \in \mathbb{R}$ is some constant.

Michael Hardy
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Adam I.
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    No. Take $d=1$, $a_1=1$, $a_2=0$, $c=0$, and check convexity for large positive $w$. –  Sep 02 '14 at 16:04

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