B.4 Derivations of Equation ( 10 ) and ( 11 )
The matrix g(F) is real symmetric since

g(F)^{\top}=\left(\frac{1}{\|F^{\top}F\|_{\mathrm{F}}+\epsilon_{F}}F^{\top}F\right)^{\top}=\frac{1}{\|F^{\top}F\|_{\mathrm{F}}+\epsilon_{F}}F^{\top}F=g(F).

Since S is also real symmetric, we can use eigendecomposition
of g(F) and S as

g(F)=Q_{F}\Lambda_{F}Q_{F}^{\top}

S=Q_{S}\Lambda_{S}Q_{S}^{\top}

where Q_{S} and Q_{F} are unitary matrices. Thus,

[S\otimes g(F)]=[Q_{S}\Lambda_{S}Q_{S}^{\top}\otimes Q_{F}\Lambda_{F}Q_{F}^{\top}]=[Q_{S}\otimes Q_{F}][\Lambda_{S}\otimes\Lambda_{F}][Q_{S}^{\top}\otimes Q_{F}^{\top}].

I_{mn}=[I_{n}\otimes I_{m}]=[Q_{S}Q_{S}^{\top}\otimes Q_{F}Q_{F}^{\top}]=[Q_{S}\otimes Q_{F}][I_{n}\otimes I_{m}][Q_{S}^{\top}\otimes Q_{F}^{\top}].

Combining,

 U =I_{mn}-\gamma[S\otimes g(F)] =[Q_{S}\otimes Q_{F}][I_{n}\otimes I_{m}][Q_{S}^{\top}\otimes Q_{F}^{\top}]-\gamma[Q_{S}\otimes Q_{F}][\Lambda_{S}\otimes\Lambda_{F}][Q_{S}^{\top}\otimes Q_{F}^{\top}] =[Q_{S}\otimes Q_{F}]([I_{n}\otimes I_{m}]-\gamma[\Lambda_{S}\otimes\Lambda_{F}])[Q_{S}^{\top}\otimes Q_{F}^{\top}].

Thus,

 U^{-1} =[Q_{S}\otimes Q_{F}]([I_{n}\otimes I_{m}]-\gamma[\Lambda_{S}\otimes\Lambda_{F}])^{-1}[Q_{S}^{\top}\otimes Q_{F}^{\top}] =[Q_{S}\otimes Q_{F}]\mathop{\mathrm{diag}}(\operatorname{vec}[G])[Q_{S}^{\top}\otimes Q_{F}^{\top}],

where the matrix G\in\mathbb{R}^{m\times n} is defined by G_{ij}=[([I_{n}\otimes I_{m}]-\gamma[\Lambda_{S}\otimes\Lambda_{F}])^{-1}]_{(i+(j-1)m)(i+(j-1)m)} (so that \operatorname{vec}[G]_{i}=[([I_{n}\otimes I_{m}]-\gamma[\Lambda_{S}\otimes\Lambda_{F}])^{-1}]_{ii} for i=1,2,\dots,mn ). By the definition of Kronecker product, this is equivalent to G_{ij}=1/(1-\gamma(\bar{\Lambda}_{F}\bar{\Lambda}_{S}^{\top})_{ij}) .

Using this form of U^{-1} ,

\operatorname{vec}[f(X,F,B)] =\operatorname{vec}\left[B\left(\lim_{H\rightarrow\infty}Z^{(H)}\right)\right] =[I_{n}\otimes B]U^{-1}\operatorname{vec}[X] =[I_{n}\otimes B][Q_{S}\otimes Q_{F}]\mathop{\mathrm{diag}}(\operatorname{vec}[G])[Q_{S}^{\top}\otimes Q_{F}^{\top}]\operatorname{vec}[X] =[Q_{S}\otimes BQ_{F}]\mathop{\mathrm{diag}}(\operatorname{vec}[G])\operatorname{vec}[Q_{F}^{\top}XQ_{S}] =[Q_{S}\otimes BQ_{F}](\operatorname{vec}[G]\circ\operatorname{vec}[Q_{F}^{\top}XQ_{S}]) =[Q_{S}\otimes BQ_{F}]\operatorname{vec}[G\circ(Q_{F}^{\top}XQ_{S})] =\operatorname{vec}[BQ_{F}(G\circ(Q_{F}^{\top}XQ_{S}))Q_{S}^{\top}]\in\mathbb{R}^{m_{y}n}

Therefore,

\mathbf{f}_{X,F,B}=f(X,F,B)=BQ_{F}(G\circ(Q_{F}^{\top}XQ_{S}))Q_{S}^{\top}\in\mathbb{R}^{m_{y}\times n}.

Similarly,

\operatorname{vec}[Z_{X,F}]=\left(\lim_{H\rightarrow\infty}Z^{(H)}\right)=U^{-1}\operatorname{vec}[X] =[Q_{S}\otimes Q_{F}]\mathop{\mathrm{diag}}(\operatorname{vec}[G])[Q_{S}^{\top}\otimes Q_{F}^{\top}]\operatorname{vec}[X] =[Q_{S}\otimes Q_{F}]\operatorname{vec}[G\circ(Q_{F}^{\top}XQ_{S})] =\operatorname{vec}[Q_{F}(G\circ(Q_{F}^{\top}XQ_{S}))Q_{S}^{\top}]\in\mathbb{R}^{mn}

Therefore,

Z_{X,F}=Q_{F}(G\circ(Q_{F}^{\top}XQ_{S}))Q_{S}^{\top}\in\mathbb{R}^{m\times n}.

Moreover, using the form of U^{-1} ,

\frac{\partial L(B,F)}{\partial\operatorname{vec}[F]} =\gamma\frac{\partial\ell_{Y}(\mathbf{f}_{X,F,B})}{\partial\operatorname{vec}[\mathbf{f}_{X,F,B}]}[I_{n}\otimes B]U^{-1}\left[SZ_{X,F}^{\top}\otimes I_{m}\right]\frac{\partial\operatorname{vec}[g(F)]}{\partial\operatorname{vec}[F]} =\gamma\frac{\partial\ell_{Y}(\mathbf{f}_{X,F,B})}{\partial\operatorname{vec}[\mathbf{f}_{X,F,B}]}[I_{n}\otimes B][Q_{S}\otimes Q_{F}]\mathop{\mathrm{diag}}(\operatorname{vec}[G])[Q_{S}^{\top}\otimes Q_{F}^{\top}]\left[SZ_{X,F}^{\top}\otimes I_{m}\right]\frac{\partial\operatorname{vec}[g(F)]}{\partial\operatorname{vec}[F]} =\gamma\frac{\partial\ell_{Y}(\mathbf{f}_{X,F,B})}{\partial\operatorname{vec}[\mathbf{f}_{X,F,B}]}[Q_{S}\otimes BQ_{F}]\mathop{\mathrm{diag}}(\operatorname{vec}[G])\left[Q_{S}^{\top}SZ_{X,F}^{\top}\otimes Q_{F}^{\top}\right]\frac{\partial\operatorname{vec}[g(F)]}{\partial\operatorname{vec}[F]}.

Thus,

 abla_{\operatorname{vec}[F]}L(B,F) =\left(\frac{\partial L(B,F)}{\partial\operatorname{vec}[F]}\right)^{\top} =\gamma\left(\frac{\partial\operatorname{vec}[g(F)]}{\partial\operatorname{vec}[F]}\right)^{\top}\left[Q_{S}^{\top}SZ_{X,F}^{\top}\otimes Q_{F}^{\top}]^{\top}\mathop{\mathrm{diag}}(\operatorname{vec}[G])]\right[Q_{S}\otimes BQ_{F}]^{\top}\left(\frac{\partial\ell_{Y}(\mathbf{f}_{X,F,B})}{\partial\operatorname{vec}[\mathbf{f}_{X,F,B}]}\right)^{\top} =\gamma\left(\frac{\partial\operatorname{vec}[g(F)]}{\partial\operatorname{vec}[F]}\right)^{\top}\left[Z_{X,F}SQ_{S}\otimes Q_{F}]\mathop{\mathrm{diag}}(\operatorname{vec}[G])]\right[Q_{S}^{\top}\otimes Q_{F}^{\top}B^{\top}]\operatorname{vec}\left[\frac{\partial\ell_{Y}(\mathbf{f}_{X,F,B})}{\partial\mathbf{f}_{X,F,B}}\right] =\gamma\left(\frac{\partial\operatorname{vec}[g(F)]}{\partial\operatorname{vec}[F]}\right)^{\top}[Z_{X,F}SQ_{S}\otimes Q_{F}]\mathop{\mathrm{diag}}(\operatorname{vec}[G])]\operatorname{vec}\left[Q_{F}^{\top}B^{\top}\frac{\partial\ell_{Y}(\mathbf{f}_{X,F,B})}{\partial\mathbf{f}_{X,F,B}}Q_{S}\right] =\gamma\left(\frac{\partial\operatorname{vec}[g(F)]}{\partial\operatorname{vec}[F]}\right)^{\top}[Z_{X,F}SQ_{S}\otimes Q_{F}]\operatorname{vec}\left[G\circ\left(Q_{F}^{\top}B^{\top}\frac{\partial\ell_{Y}(\mathbf{f}_{X,F,B})}{\partial\mathbf{f}_{X,F,B}}Q_{S}\right)\right] =\gamma\left(\frac{\partial\operatorname{vec}[g(F)]}{\partial\operatorname{vec}[F]}\right)^{\top}\operatorname{vec}\left[Q_{F}\left(G\circ\left(Q_{F}^{\top}B^{\top}\frac{\partial\ell_{Y}(\mathbf{f}_{X,F,B})}{\partial\mathbf{f}_{X,F,B}}Q_{S}\right)\right)Q_{S}^{\top}SZ_{X,F}^{\top}\right]\in\mathbb{R}^{mm}