$$
\begin{array}{l} \underset {\pi = \left\{\boldsymbol {P} ^ {(\ell)} \right\}} {\arg \min } \sum_ {\ell = 1} ^ {L} \left\| \boldsymbol {P} ^ {(\ell)} \boldsymbol {W} _ {B} ^ {(\ell)} \boldsymbol {P} ^ {(\ell - 1) ^ {\top}} \boldsymbol {H} _ {A} ^ {(\ell - 1)} - \boldsymbol {P} ^ {(\ell)} \boldsymbol {W} _ {B} ^ {(\ell)} \boldsymbol {H} _ {B} ^ {(\ell - 1)} \right\| _ {F} ^ {2} \\ \Longleftrightarrow \underset {\pi = \{\boldsymbol {P} ^ {(\ell)} \}} {\arg \min } \sum_ {\ell = 1} ^ {L} \left\| \boldsymbol {W} _ {B} ^ {(\ell)} \boldsymbol {P} ^ {(\ell - 1) ^ {\top}} \boldsymbol {H} _ {A} ^ {(\ell - 1)} - \boldsymbol {W} _ {B} ^ {(\ell)} \boldsymbol {H} _ {B} ^ {(\ell - 1)} \right\| _ {F} ^ {2} \\ \Longleftrightarrow \underset {\pi = \left\{\boldsymbol {P} ^ {(\ell)} \right\}} {\arg \min } \sum_ {\ell = 1} ^ {L} \operatorname {t r} \left(\left(\boldsymbol {H} _ {A} ^ {(\ell - 1) ^ {\top}} \boldsymbol {P} ^ {(\ell - 1)} \boldsymbol {W} _ {B} ^ {(\ell) ^ {\top}} - \boldsymbol {H} _ {B} ^ {(\ell - 1) ^ {\top}} \boldsymbol {W} _ {B} ^ {(\ell) ^ {\top}}\right) \left(\boldsymbol {W} _ {B} ^ {(\ell)} \boldsymbol {P} ^ {(\ell - 1) ^ {\top}} \boldsymbol {H} _ {A} ^ {(\ell - 1)} - \boldsymbol {W} _ {B} ^ {(\ell)} \boldsymbol {H} _ {B} ^ {(\ell - 1)}\right)\right) \\ \Longleftrightarrow \underset {\pi = \{\boldsymbol {P} ^ {(\ell)} \}} {\arg \min } \sum_ {\ell = 1} ^ {L} \operatorname {t r} \left(\boldsymbol {H} _ {A} ^ {(\ell - 1) ^ {\top}} \boldsymbol {P} ^ {(\ell - 1)} \boldsymbol {W} _ {B} ^ {(\ell) ^ {\top}} \boldsymbol {W} _ {B} ^ {(\ell)} \boldsymbol {P} ^ {(\ell - 1) ^ {\top}} \boldsymbol {H} _ {A} ^ {(\ell - 1)} - \boldsymbol {H} _ {A} ^ {(\ell - 1) ^ {\top}} \boldsymbol {P} ^ {(\ell - 1)} \boldsymbol {W} _ {B} ^ {(\ell) ^ {\top}} \boldsymbol {W} _ {B} ^ {(\ell)} \boldsymbol {H} _ {B} ^ {(\ell - 1)} \right. \\ \left. - \boldsymbol {H} _ {B} ^ {(\ell - 1) ^ {\top}} \boldsymbol {W} _ {B} ^ {(\ell)} \boldsymbol {P} ^ {(\ell - 1) ^ {\top}} \boldsymbol {H} _ {A} ^ {(\ell - 1)} + \boldsymbol {H} _ {B} ^ {(\ell - 1) ^ {\top}} \boldsymbol {W} _ {B} ^ {(\ell)} \boldsymbol {H} _ {B} ^ {(\ell - 1)}\right) \\ \Longleftrightarrow \underset {\pi = \left\{\boldsymbol {P} ^ {(\ell)} \right\}} {\arg \min } \sum_ {\ell = 1} ^ {L} \operatorname {t r} \left(\boldsymbol {H} _ {A} ^ {(\ell - 1) ^ {\top}} \boldsymbol {P} ^ {(\ell - 1)} \boldsymbol {W} _ {B} ^ {(\ell)} ^ {\top} \boldsymbol {W} _ {B} ^ {(\ell)} \boldsymbol {P} ^ {(\ell - 1) ^ {\top}} \boldsymbol {H} _ {A} ^ {(\ell - 1)} - 2 \boldsymbol {H} _ {A} ^ {(\ell - 1) ^ {\top}} \boldsymbol {P} ^ {(\ell - 1)} \boldsymbol {W} _ {B} ^ {(\ell)} ^ {\top} \boldsymbol {W} _ {B} ^ {(\ell)} \boldsymbol {H} _ {B} ^ {(\ell - 1)}\right) \\ \Longleftrightarrow \underset {\pi = \left\{\boldsymbol {P} ^ {(\ell)} \right\}} {\arg \min } \sum_ {\ell = 1} ^ {L} \operatorname {t r} \left(\boldsymbol {P} ^ {(\ell - 1)} \boldsymbol {W} _ {B} ^ {(\ell)} ^ {\top} \boldsymbol {W} _ {B} ^ {(\ell)} \boldsymbol {P} ^ {(\ell - 1)} ^ {\top} \boldsymbol {H} _ {A} ^ {(\ell - 1)} \boldsymbol {H} _ {A} ^ {(\ell - 1)} ^ {\top} - 2 \boldsymbol {P} ^ {(\ell - 1)} \boldsymbol {W} _ {B} ^ {(\ell)} ^ {\top} \boldsymbol {W} _ {B} ^ {(\ell)} \boldsymbol {H} _ {B} ^ {(\ell - 1)} \boldsymbol {H} _ {A} ^ {(\ell - 1)} ^ {\top}\right) \\ \Longleftrightarrow \underset {\pi = \{\boldsymbol {P} ^ {(\ell)} \}} {\arg \min } \sum_ {\ell = 1} ^ {L} \left(\operatorname {t r} \left(\boldsymbol {P} ^ {(\ell - 1)} \boldsymbol {W} _ {B} ^ {(\ell)} ^ {\top} \boldsymbol {W} _ {B} ^ {(\ell)} \boldsymbol {P} ^ {(\ell - 1)} ^ {\top} \boldsymbol {H} _ {A} ^ {(\ell - 1)} \boldsymbol {H} _ {A} ^ {(\ell - 1)} ^ {\top}\right) - 2 \operatorname {t r} \left(\boldsymbol {P} ^ {(\ell - 1)} \boldsymbol {W} _ {B} ^ {(\ell)} ^ {\top} \boldsymbol {W} _ {B} ^ {(\ell)} \boldsymbol {H} _ {B} ^ {(\ell - 1)} \boldsymbol {H} _ {A} ^ {(\ell - 1)} ^ {\top}\right)\right), \\ \end{array}
$$