Appendix 1: Gradient computation
The calculation of the gradient of (13) in the hermitian case, formulated in (15), is detailed below. First, using the definition of the Frobenius norm
$$\begin{aligned} \Vert {\mathbf {M}}\Vert = \left( \mathsf {trace}\left\{ {\mathbf {M}}^H{\mathbf {M}}\right\} \right) ^{\frac{1}{2}} = \left( \sum _{i,j} |M_{i,j}|^2 \right) ^{\frac{1}{2}} \end{aligned}$$
(45)
we can rewrite the criterion
$$\begin{aligned} {\mathcal {J}}_{h} ( {\mathbf {Z}}) = \sum _{i=1}^{N} \mathsf {trace}\left\{ \left( {\mathbf {U}}_{i}^{H} \right) ^H\mathsf {ODiag} \left\{ {\mathbf {U}}_{i}^{H} \right\} \right\} \end{aligned}$$
(46)
One can develop the following matrix product
$$\begin{aligned} {\mathbf {U}}_{i}^{H} = {\mathbf {T}}_i + {\mathbf {Z}}{\mathbf {T}}_i + {\mathbf {T}}_i{\mathbf {Z}}^H + {\mathbf {Z}}{\mathbf {T}}_i{\mathbf {Z}}^H \end{aligned}$$
(47)
We now compute the partial derivatives of all terms, and we only conserve those which are composed of \({\mathbf {Z}}^*\) and \({\mathbf {Z}}^H\), since the other partial derivatives are null
$$\begin{aligned} \begin{array}{ll} \partial \mathsf {trace}\left\{ \left( {\mathbf {T}}_i + {\mathbf {T}}_i{\mathbf {Z}}^H \right) ^H \mathsf {ODiag}\left\{ {\mathbf {T}}_i{\mathbf {Z}}^H \right\} \right\} &{} \partial \mathsf {trace}\left\{ \left( {\mathbf {T}}_i + {\mathbf {T}}_i{\mathbf {Z}}^H \right) ^H \mathsf {ODiag}\left\{ {\mathbf {Z}}{\mathbf {T}}_i{\mathbf {Z}}^H \right\} \right\} \\ \partial \mathsf {trace}\left\{ \left( {\mathbf {Z}}{\mathbf {T}}_i \right) ^H \mathsf {ODiag}\left\{ {\mathbf {T}}_i + {\mathbf {Z}}{\mathbf {T}}_i \right\} \right\} &{} \partial \mathsf {trace}\left\{ \left( {\mathbf {Z}}{\mathbf {T}}_i{\mathbf {Z}}^H \right) ^H \mathsf {ODiag}\left\{ {\mathbf {T}}_i + {\mathbf {Z}}{\mathbf {T}}_i \right\} \right\} \\ \partial \mathsf {trace}\left\{ \left( {\mathbf {Z}}{\mathbf {T}}_i \right) ^H \mathsf {ODiag}\left\{ {\mathbf {T}}_i{\mathbf {Z}}^H \right\} \right\} &{} \partial \mathsf {trace}\left\{ \left( {\mathbf {Z}}{\mathbf {T}}_i \right) ^H \mathsf {ODiag}\left\{ {\mathbf {Z}}{\mathbf {T}}_i{\mathbf {Z}}^H \right\} \right\} \\ \partial \mathsf {trace}\left\{ \left( {\mathbf {Z}}{\mathbf {T}}_i{\mathbf {Z}}^H \right) ^H \mathsf {ODiag}\left\{ {\mathbf {T}}_i{\mathbf {Z}}^H \right\} \right\} &{} \partial \mathsf {trace}\left\{ \left( {\mathbf {Z}}{\mathbf {T}}_i{\mathbf {Z}}^H \right) ^H \mathsf {ODiag}\left\{ {\mathbf {Z}}{\mathbf {T}}_i{\mathbf {Z}}^H \right\} \right\} \end{array} \end{aligned}$$
Applying the following property
$$\begin{aligned} \mathsf {trace}\left\{ {\mathbf {M}}\mathsf {ODiag}\left\{ {\mathbf {Q}}\right\} \right\} = \mathsf {trace}\left\{ \mathsf {ODiag}\left\{ {\mathbf {M}}\right\} {\mathbf {Q}}\right\} \end{aligned}$$
(48)
to the first terms over, and also applying
$$\begin{aligned} \mathsf {tr}\left\{ {\mathbf {M}}{\mathbf {N}}{\mathbf {Q}}\right\} = \mathsf {tr}\left\{ {\mathbf {Q}}{\mathbf {M}}{\mathbf {N}}\right\} = \mathsf {tr}\left\{ {\mathbf {N}}{\mathbf {Q}}{\mathbf {M}}\right\} \end{aligned}$$
(49)
(where \({\mathbf {M}}\), \({\mathbf {N}}\) and \({\mathbf {Q}}\) are square matrices) we get
$$\begin{aligned} \mathsf {trace}\left\{ \left( {\mathbf {T}}_i + {\mathbf {T}}_i{\mathbf {Z}}^H \right) ^H \partial \mathsf {ODiag}\left\{ {\mathbf {T}}_i{\mathbf {Z}}^H\right\} \right\}&= \mathsf {trace}\left\{ \mathsf {ODiag}\left\{ {\mathbf {T}}_i + {\mathbf {T}}_i{\mathbf {Z}}^H \right\} ^H {\mathbf {T}}_i \partial {\mathbf {Z}}^H \right\} \nonumber \\&= \mathsf {trace}\left\{ \partial {\mathbf {Z}}^H \mathsf {ODiag}\left\{ {\mathbf {T}}_i + {\mathbf {T}}_i{\mathbf {Z}}^H \right\} ^H {\mathbf {T}}_i \right\} \end{aligned}$$
(50)
Finally, the definition
$$\begin{aligned} \frac{\partial \mathsf {trace}\left\{ {\mathbf {Z}}^H {\mathbf {M}}\right\} }{\partial {\mathbf {Z}}^*} = {\mathbf {M}} \end{aligned}$$
(51)
applied here leads to
$$\begin{aligned} \frac{\partial \mathsf {trace}\left\{ {\mathbf {Z}}^H \mathsf {ODiag}\left\{ {\mathbf {T}}_i + {\mathbf {T}}_i{\mathbf {Z}}^H \right\} ^H {\mathbf {T}}_i \right\} }{\partial {\mathbf {Z}}^*} = \mathsf {ODiag}\left\{ {\mathbf {T}}_i + {\mathbf {T}}_i{\mathbf {Z}}^H \right\} ^H {\mathbf {T}}_i \end{aligned}$$
(52)
Doing this to all the terms to derivate gives
$$\begin{aligned} \frac{\partial \mathsf {trace}\left\{ \left( {\mathbf {T}}_i + {\mathbf {T}}_i{\mathbf {Z}}^H \right) ^H \mathsf {ODiag}\left\{ {\mathbf {Z}}{\mathbf {T}}_i{\mathbf {Z}}^H\right\} \right\} }{\partial {\mathbf {Z}}^*}&= \mathsf {ODiag}\left\{ {\mathbf {T}}_i + {\mathbf {T}}_i{\mathbf {Z}}^H \right\} ^H{\mathbf {Z}}{\mathbf {T}}_i \nonumber \\ \frac{\partial \mathsf {trace}\left\{ \left( {\mathbf {Z}}{\mathbf {T}}_i{\mathbf {Z}}^H\right) ^H\mathsf {ODiag}\left\{ {\mathbf {T}}_i + {\mathbf {Z}}{\mathbf {T}}_i \right\} \right\} }{\partial {\mathbf {Z}}^*}&= \mathsf {ODiag}\left\{ {\mathbf {T}}_i + {\mathbf {Z}}{\mathbf {T}}_i \right\} {\mathbf {Z}}{\mathbf {T}}_i^H \nonumber \\ \frac{\partial \mathsf {trace}\left\{ \left( {\mathbf {Z}}{\mathbf {T}}_i\right) ^H\mathsf {ODiag}\left\{ {\mathbf {T}}_i{\mathbf {Z}}^H \right\} \right\} }{\partial {\mathbf {Z}}^*}&= \mathsf {ODiag}\left\{ {\mathbf {T}}_i{\mathbf {Z}}^H \right\} {\mathbf {T}}_i^H + \mathsf {ODiag}\left\{ {\mathbf {Z}}{\mathbf {T}}_i \right\} ^H{\mathbf {T}}_i \nonumber \\ \frac{\partial \mathsf {trace}\left\{ \left( {\mathbf {Z}}{\mathbf {T}}_i\right) ^H\mathsf {ODiag}\left\{ {\mathbf {Z}}{\mathbf {T}}_i{\mathbf {Z}}^H \right\} \right\} }{\partial {\mathbf {Z}}^*}&= \mathsf {ODiag}\left\{ {\mathbf {Z}}{\mathbf {T}}_i{\mathbf {Z}}^H \right\} {\mathbf {T}}_i^H + \mathsf {ODiag}\left\{ {\mathbf {Z}}{\mathbf {T}}_i \right\} ^H{\mathbf {Z}}{\mathbf {T}}_i \nonumber \\ \frac{\partial \mathsf {trace}\left\{ \left( {\mathbf {Z}}{\mathbf {T}}_i{\mathbf {Z}}^H\right) ^H \mathsf {ODiag}\left\{ {\mathbf {T}}_i{\mathbf {Z}}^H\right\} \right\} }{\partial {\mathbf {Z}}^*}&= \mathsf {ODiag}\left\{ {\mathbf {T}}_i{\mathbf {Z}}^H \right\} {\mathbf {Z}}{\mathbf {T}}_i^H \nonumber \\&\quad + \mathsf {ODiag}\left\{ {\mathbf {Z}}{\mathbf {T}}_i{\mathbf {Z}}^H \right\} ^H {\mathbf {T}}_i \nonumber \\ \frac{\partial \mathsf {trace}\left\{ \left( {\mathbf {Z}}{\mathbf {T}}_i{\mathbf {Z}}^H\right) ^H\mathsf {ODiag}\left\{ {\mathbf {Z}}{\mathbf {T}}_i{\mathbf {Z}}^H \right\} \right\} }{\partial {\mathbf {Z}}^*}&= \mathsf {ODiag}\left\{ {\mathbf {Z}}{\mathbf {T}}_i{\mathbf {Z}}^H \right\} {\mathbf {Z}}{\mathbf {T}}_i^H \nonumber \\&\quad + \mathsf {ODiag}\left\{ {\mathbf {Z}}{\mathbf {T}}_i{\mathbf {Z}}^H \right\} ^H{\mathbf {Z}}{\mathbf {T}}_i\nonumber \\ \frac{\partial \mathsf {trace}\left\{ \left( {\mathbf {Z}}{\mathbf {T}}_i\right) ^H \mathsf {ODiag}\left\{ {\mathbf {T}}_i + {\mathbf {Z}}{\mathbf {T}}_i \right\} \right\} }{\partial {\mathbf {Z}}^*}&= \mathsf {ODiag}\left\{ {\mathbf {T}}_i + {\mathbf {Z}}{\mathbf {T}}_i \right\} {\mathbf {T}}_i^H \end{aligned}$$
(53)
and we then get all the terms composing the gradient of the criterion.
Appendix 2: Computation of the optimal stepsize for the initial criterion
We first remind the formulation of the criterion (13) :
$$\begin{aligned} {\mathcal {J}} ( {\mathbf {Z}}) = \sum _{i=1}^{N} \Vert \mathsf {ODiag}\{ \left( {\mathbf {I}}+ {\mathbf {Z}}\right) {\mathbf {T}}_{i} \left( {\mathbf {I}}+ {\mathbf {Z}}\right) ^{\ddagger } \} \Vert ^2 \end{aligned}$$
(54)
and we use the update of \({\mathbf {Z}}\) in (14)
$$\begin{aligned} {\mathbf {Z}}= - \mu \frac{\partial {\mathcal {J}}( {\mathbf {Z}}) }{\partial {\mathbf {Z}}^*} = \mu {\mathbf {F}} \end{aligned}$$
(55)
We then introduce (55) in (54), and get
$$\begin{aligned} {\mathcal {J}} ( {\mathbf {Z}}) = \sum _{i=1}^{N}\Vert \mathsf {ODiag}\{ \left( {\mathbf {I}}+ \mu {\mathbf {F}}\right) {\mathbf {T}}_{i} \left( {\mathbf {I}}+ \mu {\mathbf {F}}\right) ^{\ddagger } \} \Vert ^2 \end{aligned}$$
(56)
Using the definitions (45) and (48) into the equation above leads to
$$\begin{aligned} {\mathcal {J}} ( {\mathbf {Z}})&= \sum _{i=1}^{N} \mathsf {tr}\left\{ \left( \left( {\mathbf {I}}+ \mu {\mathbf {F}}\right) {\mathbf {T}}_{i} \left( {\mathbf {I}}+ \mu {\mathbf {F}}\right) ^{\ddagger }\right) ^H \right. \nonumber \\&\quad \, \times \left. \mathsf {ODiag}\{\left( {\mathbf {I}}+ \mu {\mathbf {F}}\right) {\mathbf {T}}_{i} \left( {\mathbf {I}}+ \mu {\mathbf {F}}\right) ^{\ddagger } \} \right\} \end{aligned}$$
(57)
The development of \(\left( {\mathbf {I}}+ \mu {\mathbf {F}}\right) {\mathbf {T}}_{i} \left( {\mathbf {I}}+ \mu {\mathbf {F}}\right) ^{\ddagger }\) gives a second order polynomial in \(\mu \):
$$\begin{aligned} \left( {\mathbf {I}}+ \mu {\mathbf {F}}\right) {\mathbf {T}}_{i} \left( {\mathbf {I}}+ \mu {\mathbf {F}}\right) ^{\ddagger } = {\mathbf {T}}_{i} + \mu \left( {\mathbf {F}}{\mathbf {T}}_{i} + {\mathbf {T}}_{i}{\mathbf {F}}^{\ddagger }\right) + \mu ^2{\mathbf {F}}{\mathbf {T}}_{i}{\mathbf {F}}^{\ddagger } \end{aligned}$$
(58)
Finally, developping the matrix product in argument of the trace function leads to a fourth order polynomial \(\mu \) (36), where the coefficients are given below
$$\begin{aligned} J_0&= {\mathbf {T}}_i^H \mathsf {ODiag}\{{\mathbf {T}}_i\} \nonumber \\ J_1&= {\mathbf {T}}_i^H \mathsf {ODiag}\{{\mathbf {F}}{\mathbf {T}}_i + {\mathbf {T}}_i{\mathbf {F}}^\ddagger \} + \left( {\mathbf {F}}{\mathbf {T}}_i + {\mathbf {T}}_i{\mathbf {F}}^\ddagger \right) ^H\mathsf {ODiag}\{{\mathbf {T}}_i\} \nonumber \\ J_2&= {\mathbf {T}}_i^H \mathsf {ODiag}\{{\mathbf {F}}{\mathbf {T}}_i{\mathbf {F}}^\ddagger \} + \left( {\mathbf {F}}{\mathbf {T}}_i{\mathbf {F}}^\ddagger \right) ^H\mathsf {ODiag}\{{\mathbf {T}}_i\} + \left( {\mathbf {F}}{\mathbf {T}}_i + {\mathbf {T}}_i{\mathbf {F}}^\ddagger \right) ^H\mathsf {ODiag}\{{\mathbf {F}}{\mathbf {T}}_i + {\mathbf {T}}_i{\mathbf {F}}^\ddagger \}\nonumber \\ J_3&= \left( {\mathbf {F}}{\mathbf {T}}_i{\mathbf {F}}^\ddagger \right) ^H\mathsf {ODiag}\{{\mathbf {F}}{\mathbf {T}}_i + {\mathbf {T}}_i{\mathbf {F}}^\ddagger \} + \left( {\mathbf {F}}{\mathbf {T}}_i + {\mathbf {T}}_i{\mathbf {F}}^\ddagger \right) ^H\mathsf {ODiag}\{{\mathbf {F}}{\mathbf {T}}_i{\mathbf {F}}^\ddagger \}\nonumber \\ J_4&= \left( {\mathbf {F}}{\mathbf {T}}_i{\mathbf {F}}^\ddagger \right) ^H\mathsf {ODiag}\{{\mathbf {F}}{\mathbf {T}}_i{\mathbf {F}}^\ddagger \} \end{aligned}$$
(59)
Appendix 3: Computation of the gradient for approximated criteria
We develop here the computation of the gradient of the criterion (28) in the hermitian case, where the result is done in (32). Applying (45) to this criterion (with \(\mathbf {E}_i=\mathbf {T}_i^{(1)}\) and \(\mathbf {F}_i=\mathbf {T}_i^{(2)})\)
$$\begin{aligned} \mathcal {J}_{a,h} ( {\mathbf {Z}}) = \sum _{i=1}^{N} \mathsf {trace}\left\{ \left( {\mathbf {T}}_{i}^{(1)} + {\mathbf {Z}}{\mathbf {T}}_{i}^{(2)} + {\mathbf {T}}_{i}^{(2)}{\mathbf {Z}}^{H} \right) ^H \mathsf {ODiag}\left\{ {\mathbf {T}}_{i}^{(1)} + {\mathbf {Z}}{\mathbf {T}}_{i}^{(2)} + {\mathbf {T}}_{i}^{(2)}{\mathbf {Z}}^{H} \right\} \right\} \end{aligned}$$
(60)
Keeping all the partial derivatives composed of \({\mathbf {Z}}^*\) et \({\mathbf {Z}}^H\):
$$\begin{aligned} \partial&\mathsf {trace}\left\{ \left( {\mathbf {T}}_{i}^{(1)} + {\mathbf {T}}_{i}^{(2)}{\mathbf {Z}}^{H} \right) ^H \mathsf {ODiag}\left\{ {\mathbf {T}}_{i}^{(2)}{\mathbf {Z}}^{H} \right\} \right\} \qquad \partial \mathsf {trace}\left\{ \left( {\mathbf {Z}}{\mathbf {T}}_{i}^{(2)} \right) ^H \mathsf {ODiag}\left\{ {\mathbf {T}}_{i}^{(1)} + {\mathbf {Z}}{\mathbf {T}}_{i}^{(2)} \right\} \right\} \\ \partial&\mathsf {trace}\left\{ \left( {\mathbf {Z}}{\mathbf {T}}_{i}^{(2)} \right) ^H \mathsf {ODiag}\left\{ {\mathbf {T}}_{i}^{(2)}{\mathbf {Z}}^{H} \right\} \right\} \end{aligned}$$
Applying (48), (49) and (51) to the terms above
$$\begin{aligned} \frac{\partial \mathsf {trace}\left\{ \left( {\mathbf {Z}}{\mathbf {T}}_{i}^{(2)} \right) ^H \mathsf {ODiag}\left\{ {\mathbf {T}}_{i}^{(2)}{\mathbf {Z}}^{H} \right\} \right\} }{\partial {\mathbf {Z}}^*}&= \mathsf {ODiag}\left\{ {\mathbf {T}}_{i}^{(2)}{\mathbf {Z}}^{H}\right\} \left( {\mathbf {T}}_{i}^{(2)}\right) ^H \nonumber \\&\quad + \mathsf {ODiag}\left\{ {\mathbf {Z}}{\mathbf {T}}_{i}^{(2)}\right\} ^H{\mathbf {T}}_{i}^{(2)} \nonumber \\ \frac{\partial \mathsf {trace}\left\{ \left( {\mathbf {Z}}{\mathbf {T}}_{i}^{(2)} \right) ^H \mathsf {ODiag}\left\{ {\mathbf {T}}_{i}^{(1)} + {\mathbf {Z}}{\mathbf {T}}_{i}^{(2)} \right\} \right\} }{\partial {\mathbf {Z}}^*}&= \mathsf {ODiag}\left\{ {\mathbf {T}}_{i}^{(1)} + {\mathbf {Z}}{\mathbf {T}}_{i}^{(2)} \right\} \left( {\mathbf {T}}_{i}^{(2)}\right) ^H \nonumber \\ \frac{\partial \mathsf {trace}\left\{ \left( {\mathbf {T}}_{i}^{(1)} + {\mathbf {T}}_{i}^{(2)}{\mathbf {Z}}^{H} \right) ^H \mathsf {ODiag}\left\{ {\mathbf {T}}_{i}^{(2)}{\mathbf {Z}}^{H} \right\} \right\} }{\partial {\mathbf {Z}}^*}&= \mathsf {ODiag}\left\{ {\mathbf {T}}_{i}^{(1)} + {\mathbf {T}}_{i}^{(2)}{\mathbf {Z}}^{H}\right\} ^H {\mathbf {T}}_{i}^{(2)} \end{aligned}$$
(61)
leads to the expression given in (32). The formulation of the gradient of (33) is reached by not considering the terms including \({\mathbf {Z}}\) in (61).
Appendix 4: Computation of the optimal stepsize for the approximated criterion
Once approximations have been applied to the criterion (13), we, in a first time, reach the criterion (28).
$$\begin{aligned} {\mathcal {J}}_{a} ( {\mathbf {Z}}) = \sum _{i=1}^{N} \Vert \mathsf {ODiag}\{ {\mathbf {T}}_{i}^{(1)} + {\mathbf {Z}}{\mathbf {T}}_{i}^{(2)} + {\mathbf {T}}_{i}^{(2)}{\mathbf {Z}}^{\ddagger } \} \Vert ^2 \end{aligned}$$
(62)
The principle used to get the gradient of the exact criterion is kept. Replacing \({\mathbf {Z}}\) by (55) in (62) leads to
$$\begin{aligned} {\mathcal {J}}_{a} ( {\mathbf {Z}}) = \sum _{i=1}^{N} \Vert \mathsf {ODiag}\{ {\mathbf {T}}_{i}^{(1)} + \mu {\mathbf {F}}{\mathbf {T}}_{i}^{(2)} + {\mathbf {T}}_{i}^{(2)}\mu {\mathbf {F}}^{\ddagger } \} \Vert ^2 \end{aligned}$$
(63)
Then, using (45) and (48) in the equation above, we get
$$\begin{aligned} {\mathcal {J}}_{a} ( {\mathbf {Z}}) = \sum _{i=1}^{N} \mathsf {tr}\left\{ \left( {\mathbf {T}}_{i}^{(1)} + \mu {\mathbf {F}}{\mathbf {T}}_{i}^{(2)} + {\mathbf {T}}_{i}^{(2)}\mu {\mathbf {F}}^{\ddagger } \right) ^H \mathsf {ODiag}\left\{ {\mathbf {T}}_{i}^{(1)} + \mu {\mathbf {F}}{\mathbf {T}}_{i}^{(2)} + {\mathbf {T}}_{i}^{(2)}\mu {\mathbf {F}}^{\ddagger } \right\} \right\} \end{aligned}$$
(64)
Developping the matrix product in argument of the trace function leads to the second order polynomial in \(\mu \) (37), where the coefficients are given below
$$\begin{aligned} J_{a,0}&= {{\mathbf {T}}_{i}^{(1)}}^H \mathsf {ODiag}\left\{ {\mathbf {T}}_{i}^{(1)} \right\} \nonumber \\ J_{a,1}&= {{\mathbf {T}}_{i}^{(1)}}^H \mathsf {ODiag}\left\{ {\mathbf {F}}{\mathbf {T}}_{i}^{(2)} + {\mathbf {T}}_{i}^{(2)}{\mathbf {F}}^\ddagger \right\} + \left( {\mathbf {F}}{\mathbf {T}}_{i}^{(2)} + {\mathbf {T}}_{i}^{(2)}{\mathbf {F}}^\ddagger \right) ^H\mathsf {ODiag}\left\{ {\mathbf {T}}_{i}^{(1)} \right\} \nonumber \\ J_{a,2}&= \left( {\mathbf {F}}{\mathbf {T}}_{i}^{(2)} + {\mathbf {T}}_{i}^{(2)}{\mathbf {F}}^\ddagger \right) ^H\mathsf {ODiag}\left\{ {\mathbf {F}}{\mathbf {T}}_{i}^{(2)} + {\mathbf {T}}_{i}^{(2)}{\mathbf {F}}^\ddagger \right\} \end{aligned}$$
(65)
Concerning the criterion (31), we remark that the difference comes from the approximation in \({\mathbf {Z}}\) in (64). Then, we only have a first order polynomial in \(\mu \) where the coefficients are those which do not contain a double product in \({\mathbf {Z}}\), whether \(b_{i,0}\) or \(b_{i,1}\).
