Constrained MLLR Transformation Matrix (CMLLR)

Substituting the for expressions for CMLLR adaptation where^9.6

$$\hat{{\mbox{\boldmath$\mu$}}}_{m_r} = {\mbox{\boldmath$H$}}_r{\mb... ...{{\mbox{\boldmath$\Sigma$}}}_{m_r}{{\mbox{\boldmath$H$}}}_r^{\scriptstyle\sf T}$$ (9.19)

into the auxiliary function, and using the fact that the covariance matrices are diagonal yields

$${\cal Q}({\cal M},{\hat{\cal M}}) = K + \sum_{r=1}^R\left[ \beta\... ...ptstyle\sf T}_{rj} - 2{\mbox{\boldmath$w$}}_{rj}{\bf k}^{(j)}_r \right)}\right]$$

where

$${\mbox{\boldmath$W$}}_r = \left[\begin{array}{c c} -{\mbox{\boldm... ...in{array}{c c} {\mbox{\boldmath$b$}} & {\mbox{\boldmath$A$}} \end{array}\right]$$ (9.20)

${\mbox{\boldmath $w$}}_{ri}$ is $i^{th}$ row of ${\mbox{\boldmath $W$}}_r$, the $1\times n$ row vector ${\bf p}_{ri}$ is the zero extended vector of cofactors of ${\bf A}_r$, ${\bf G}^{(i)}_r$ and ${\bf k}^{(i)}_r$ are defined as

$${\bf G}^{(i)}_r=\sum_{m_r=1}^{M_r} \frac{1}{\sigma_{m_ri}^{2}} \s... ...mbox{\boldmath$\zeta$}}}(t){{\mbox{\boldmath$\zeta$}}}^{{\scriptstyle\sf T}}(t)$$ (9.21)

and

$${\bf k}^{(i)}_r=\sum_{m_r=1}^{M_r} \frac{\mu_{m_ri}}{\sigma_{m_ri}^{2}} \sum_{t=1}^TL_{m_r}(t){{\mbox{\boldmath$\zeta$}}}^{{\scriptstyle\sf T}}(t)$$ (9.22)

Differentiating the auxiliary function with respect to the transform ${\mbox{\boldmath $W$}}_r$ , and then maximising it with respect to the transformed mean yields the following update

$${\mbox{\boldmath$w$}}_{ri} = \left(\alpha{{\bf p}_{ri}} + {\bf k}^{(i)}_r\right){\bf G}^{(i)-1}_r$$ (9.23)

where $\alpha$ satisfies

$$\alpha^2{\bf p}_{ri}{\bf G}^{(i)-1}_r{\bf p}_{ri}^{\scriptstyle\s... ...\alpha{\bf p}_{ri}{\bf G}^{(i)-1}_r{\bf k}^{(i){\scriptstyle\sf T}}_r - \beta=0$$ (9.24)

There are thus two possible solutions for $\alpha$. The solutions that yields the maximum increase in the auxiliary function (obtained by simply substituting in the two options) is used. This is an iterative optimisation scheme as the cofactors mean the estimate of row $i$ is dependent on all the other rows (in that block).