I'm trying to perform an optimization with the aim to find the rotation that minimizes the $l_1$ norm of a third order tensor $Q_{ijk}$ whose entries are known.
The rotation of the tensor is defined as $\tilde{Q}_{ijk} = \sum_{p,q,r = 1}^{N} X_{pi}Q_{pqr}X_{qj}X_{rk}$.
Since the numerical evaluation of its gradient is quite costly, I'd like to compute analytically the derivative of the rotated tensor $\tilde{Q}_{ijk}$ with respect the rotation matrix $\boldsymbol{X}$. Therefore,I'd like to find a closed expression for $\frac{\partial \boldsymbol{\tilde{Q}}}{\partial \boldsymbol{X}}$ or even better for its $l_1$ norm.
Thanks in advance for any help!
Riccardo