I'm still not very confident with tensor calculus and I came across a paper that was solving an optimization problem based on Tucker decomposition; I don't understand how, from the initial formulation, it ends up in a simplified optimization problem with respect to each-mode basis matrix.
Let $\pmb Y \in R^{I_1 \times I_2 \times I_3}$ a 3D-tensor, $\pmb G \in R^M_1 \times M_2 \times I_3$ the core tensor and $A \in R^{I_1 \times M_1} B \in R^{I_2 \times M_2}$
The initial formulation is $$ min_{(A,B,\pmb G)} ||\pmb Y - \pmb G \times_1 A \times_2 B ||_F^2 $$ subject to $ A^TA=I_{M_1}, B^TB=I_{M_2}$
The paper says "substituting one of the KKT conditions $ \pmb G= \pmb Y \times_1 A^T \times_2 B^T $ we can obtain a problem wrt each mode".
Defining $X_A:= [\pmb Y \times_2 B^T ]_{(1)} \in R^{I_1 \times M_2I_3}$ and $X_B:= [\pmb Y \times_1 A^T ]_{(2)} \in R^{I_2 \times M_1I_3}$ the problem becomes
$$min_{A} ||X_A - AA^TX_A ||_F^2 \quad AA^T=I_{M_1}$$ $$min_{B} ||X_B - BB^TX_B ||_F^2 \quad BB^T=I_{M_2}$$
I stopped to the point where I can write the initial formulation as: $$ min_{(A,B)} ||\pmb Y - \pmb Y \times_1 A A^T \times_2 BB^T ||_F^2 $$
Thank you in advance