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Question
How do I find the tensor rank decomposition of the following tensor of shape $2 \times 2 \times 2$?
\[\left[\begin{matrix}\left[\begin{matrix}\frac{\sqrt{2} \left(2 \mu - 2 x\right) e^{- \frac{\left(- \mu + x\right)^{2}}{2 \sigma^{2}}}}{4 \sqrt{\pi} \sigma^{5}} + \frac{\sqrt{2} \left(8 \mu - 8 x\right) e^{- \frac{\left(- \mu + x\right)^{2}}{2 \sigma^{2}}}}{8 \sqrt{\pi} \sigma^{5}} - \frac{\sqrt{2} \left(2 \mu - 2 x\right)^{3} e^{- \frac{\left(- \mu + x\right)^{2}}{2 \sigma^{2}}}}{16 \sqrt{\pi} \sigma^{7}} & \frac{3 \sqrt{2} e^{- \frac{\left(- \mu + x\right)^{2}}{2 \sigma^{2}}}}{2 \sqrt{\pi} \sigma^{4}} - \frac{\sqrt{2} \left(- \mu + x\right)^{2} e^{- \frac{\left(- \mu + x\right)^{2}}{2 \sigma^{2}}}}{2 \sqrt{\pi} \sigma^{6}} - \frac{5 \sqrt{2} \left(2 \mu - 2 x\right)^{2} e^{- \frac{\left(- \mu + x\right)^{2}}{2 \sigma^{2}}}}{8 \sqrt{\pi} \sigma^{6}} + \frac{\sqrt{2} \left(- \mu + x\right)^{2} \left(2 \mu - 2 x\right)^{2} e^{- \frac{\left(- \mu + x\right)^{2}}{2 \sigma^{2}}}}{8 \sqrt{\pi} \sigma^{8}}\\\frac{3 \sqrt{2} e^{- \frac{\left(- \mu + x\right)^{2}}{2 \sigma^{2}}}}{2 \sqrt{\pi} \sigma^{4}} - \frac{\sqrt{2} \left(- \mu + x\right)^{2} e^{- \frac{\left(- \mu + x\right)^{2}}{2 \sigma^{2}}}}{2 \sqrt{\pi} \sigma^{6}} - \frac{5 \sqrt{2} \left(2 \mu - 2 x\right)^{2} e^{- \frac{\left(- \mu + x\right)^{2}}{2 \sigma^{2}}}}{8 \sqrt{\pi} \sigma^{6}} + \frac{\sqrt{2} \left(- \mu + x\right)^{2} \left(2 \mu - 2 x\right)^{2} e^{- \frac{\left(- \mu + x\right)^{2}}{2 \sigma^{2}}}}{8 \sqrt{\pi} \sigma^{8}} & - \frac{3 \sqrt{2} \left(2 \mu - 2 x\right) e^{- \frac{\left(- \mu + x\right)^{2}}{2 \sigma^{2}}}}{\sqrt{\pi} \sigma^{5}} + \frac{9 \sqrt{2} \left(- \mu + x\right)^{2} \left(2 \mu - 2 x\right) e^{- \frac{\left(- \mu + x\right)^{2}}{2 \sigma^{2}}}}{4 \sqrt{\pi} \sigma^{7}} - \frac{\sqrt{2} \left(- \mu + x\right)^{4} \left(2 \mu - 2 x\right) e^{- \frac{\left(- \mu + x\right)^{2}}{2 \sigma^{2}}}}{4 \sqrt{\pi} \sigma^{9}}\end{matrix}\right] & \left[\begin{matrix}\frac{3 \sqrt{2} e^{- \frac{\left(- \mu + x\right)^{2}}{2 \sigma^{2}}}}{2 \sqrt{\pi} \sigma^{4}} - \frac{\sqrt{2} \left(- \mu + x\right)^{2} e^{- \frac{\left(- \mu + x\right)^{2}}{2 \sigma^{2}}}}{2 \sqrt{\pi} \sigma^{6}} - \frac{5 \sqrt{2} \left(2 \mu - 2 x\right)^{2} e^{- \frac{\left(- \mu + x\right)^{2}}{2 \sigma^{2}}}}{8 \sqrt{\pi} \sigma^{6}} + \frac{\sqrt{2} \left(- \mu + x\right)^{2} \left(2 \mu - 2 x\right)^{2} e^{- \frac{\left(- \mu + x\right)^{2}}{2 \sigma^{2}}}}{8 \sqrt{\pi} \sigma^{8}} & - \frac{3 \sqrt{2} \left(2 \mu - 2 x\right) e^{- \frac{\left(- \mu + x\right)^{2}}{2 \sigma^{2}}}}{\sqrt{\pi} \sigma^{5}} + \frac{9 \sqrt{2} \left(- \mu + x\right)^{2} \left(2 \mu - 2 x\right) e^{- \frac{\left(- \mu + x\right)^{2}}{2 \sigma^{2}}}}{4 \sqrt{\pi} \sigma^{7}} - \frac{\sqrt{2} \left(- \mu + x\right)^{4} \left(2 \mu - 2 x\right) e^{- \frac{\left(- \mu + x\right)^{2}}{2 \sigma^{2}}}}{4 \sqrt{\pi} \sigma^{9}}\\- \frac{3 \sqrt{2} \left(2 \mu - 2 x\right) e^{- \frac{\left(- \mu + x\right)^{2}}{2 \sigma^{2}}}}{\sqrt{\pi} \sigma^{5}} - \frac{2 \sqrt{2} \left(- \mu + x\right)^{3} e^{- \frac{\left(- \mu + x\right)^{2}}{2 \sigma^{2}}}}{\sqrt{\pi} \sigma^{7}} + \frac{5 \sqrt{2} \left(- \mu + x\right)^{2} \left(2 \mu - 2 x\right) e^{- \frac{\left(- \mu + x\right)^{2}}{2 \sigma^{2}}}}{4 \sqrt{\pi} \sigma^{7}} - \frac{\sqrt{2} \left(- \mu + x\right)^{4} \left(2 \mu - 2 x\right) e^{- \frac{\left(- \mu + x\right)^{2}}{2 \sigma^{2}}}}{4 \sqrt{\pi} \sigma^{9}} & - \frac{3 \sqrt{2} e^{- \frac{\left(- \mu + x\right)^{2}}{2 \sigma^{2}}}}{\sqrt{\pi} \sigma^{4}} + \frac{27 \sqrt{2} \left(- \mu + x\right)^{2} e^{- \frac{\left(- \mu + x\right)^{2}}{2 \sigma^{2}}}}{2 \sqrt{\pi} \sigma^{6}} - \frac{6 \sqrt{2} \left(- \mu + x\right)^{4} e^{- \frac{\left(- \mu + x\right)^{2}}{2 \sigma^{2}}}}{\sqrt{\pi} \sigma^{8}} + \frac{\sqrt{2} \left(- \mu + x\right)^{6} e^{- \frac{\left(- \mu + x\right)^{2}}{2 \sigma^{2}}}}{2 \sqrt{\pi} \sigma^{10}}\end{matrix}\right]\end{matrix}\right]\]Important: I do not know that the decomposition exists, so a disproof of existence would also suffice.
Post-Mortem
I think for a lot of people, including myself, the tensor I was considering is absurdly complicated. It was probably too much work for someone to casually solve as a problem they found on the internet, especially since it lacks any obvious motivation as a problem.