adapt.tex

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                \liksum{jm}\bm{o}^r_{t}}{\liksum{jm}}
\]

As can be seen, if the occupation likelihood
of a Gaussian component ($N_{jm}$) is small, then the
mean MAP estimate will remain close to the speaker
independent component mean. 
With MAP adaptation, every single mean
component in the system is updated with a MAP estimate, based on the
prior mean, the weighting and the adaptation data. Hence, MAP
adaptation requires a new ``speaker-dependent'' model set to be saved.

One obvious drawback to MAP adaptation is that it requires more
adaptation data to be effective when compared to MLLR, because MAP
adaptation is specifically defined at the component level. When
larger amounts of adaptation training data become available, MAP
begins to perform better than MLLR, due to this detailed update of
each component (rather than the pooled Gaussian transformation
approach of MLLR). In fact the two adaptation processes can be
combined to improve performance still further, by using the MLLR
transformed means as the priors for MAP adaptation (by replacing
$\bm{\mu}_{jm}$ in equation~\ref{e:meanmap} with the transformed mean
of equation~\ref{e:mtrans}). In this case
components that have a low occupation likelihood in the adaptation
data, (and hence would not change much using MAP alone) have been
adapted using a regression class transform in MLLR. An example usage
is shown in the following section. 

%\pagebreak

\mysect{Linear Transformation Estimation Formulae}{mllrformulae}

For reference purposes, this section lists the various formulae
employed within the \HTK\ adaptation tool\index{adaptation!MLLR
formulae}. It is assumed throughout
that single stream data is used and that diagonal covariances are also
used. All are standard and can be found in various literature.
 
The following notation is used in this section
\begin{tabbing}
++ \= ++++++++ \= \kill
\> $\mathcal{M}$ \> the model set\\
\> $\hat{\mathcal{M}}$ \> the adapted model set\\
\> $T$ \> number of observations \\
\> $m$ \> a mixture component \\
\> $\bm{O}$      \> a sequence of $d$-dimensional observations \\
\> $\bm{o}(t)$    \> the observation at time $t$, $1 \leq t \leq T $\\
\> $\bm{\zeta}(t)$\> extended observation at time $t$, $1 \leq t \leq T $
\\
\> $\bm{\mu}_{m_r}$  \> mean vector for the mixture component $m_r$\\
\> $\bm{\xi}_{m_r}$  \> extended mean vector for the mixture component $m_r$\\
\> $\bm{\Sigma}_{m_r}$  \> covariance matrix for the mixture component $m_r$ \\
\> $L_{m_r}(t)$ \> the occupancy probability for the mixture component $m_r$\\
\>              \>   at time $t$        
\end{tabbing}

To enable robust transformations to be trained, the transform matrices
are tied across a number of Gaussians. The set of Gaussians which
share a transform is referred to as a regression class.  For a
particular transform case $\bm{W_r}$, the $M_r$ Gaussian components
$\left\{m_1, m_2, \dots, m_{M_r}\right\}$ will be tied together, as
determined by the regression class tree (see
section~\ref{s:reg_classes}).  The standard auxiliary
function shown below is used to estimate the transforms.
\newcommand{\like}{L_{m_r}(t)}
\begin{eqnarray}
{\cal Q}({\cal M},{\hat{\cal M}}) = 
- \frac{1}{2}
\sum_{r=1}^R
\sum_{m_r=1}^{M_r}
\sum_{t=1}^T
\like\left[
K^{(m)}+\log(|{\hat{\bm\Sigma}}_{m_r}|)+({\bm o}(t)-
{\hat{\bm\mu}}_{m_r})^T{{\hat{\bm\Sigma}}}_{m_r}^{-1}({\bm o}(t)-{\hat{\bm\mu}}_{m_r})
\right] \nonumber
\end{eqnarray}
where $K^{(m)}$ subsumes all constants and $\like$, the occupation likelihood, is defined as,
\[ 
         \like = p(q_{m_r}(t)\;|\;\mathcal{M}, \bm{O}_T)
\]
and $q_{m_r}(t)$ indicates the Gaussian component $m_r$ at time $t$,
and $\bm{O}_T = \left\{\bm{o}(1),\dots,\bm{o}(T)\right\}$ is the
adaptation data. The occupation likelihood is obtained from the
forward-backward process described in section~\ref{s:bwformulae}.


\mysubsect{Mean Transformation Matrix ({\tt MLLRMEAN})}{mtransest}
Substituting the for expressions for MLLR mean adaptation 
\begin{eqnarray}
\hat{\bm{\mu}}_{m_r} = \bm{W}_r\bm{\xi}_{m_r}, \:\:\:\: 
\hat{\bm{\Sigma}}_{m_r} = {\bm{\Sigma}}_{m_r}
\end{eqnarray}
into the auxiliary function, and using the fact that the covariance
matrices are diagonal, yields
\begin{eqnarray}
{\cal Q}({\cal M},{\hat{\cal M}}) = K
- \frac{1}{2}
\sum_{r=1}^R
\sum_{j=1}^d{
\left({\bm{w}}_{rj}{\bf G}^{(j)}_r{\bm{w}}^T_{rj} -2 {\bm{w}}_{rj}{\bf k}^{(j)T}_r
\right)} \nonumber
\end{eqnarray}
where ${\bm{w}}_{rj}$ is the $j^{th}$ {\em row} of $\bm{W}_r$,
\begin{eqnarray}
{\bf G}^{(i)}_r=\sum_{m_r=1}^{M_r}
\frac{1}{\sigma_{m_ri}^{2}}{\bm\xi}_{m_r}{\bm\xi}^{T}_{m_r}
\sum_{t=1}^T\like
\label{eq:gi_mllr}
\end{eqnarray}
and
\begin{eqnarray}
{\bf k}^{(i)}_{r} =  \sum_{m_r=1}^{M_r}\sum\limits_{t=1}^T
\like\frac{1}{\sigma^{2}_{m_ri}} 
o_i(t){\bm\xi}^{T}_{m_r}
\end{eqnarray}
Differentiating the auxiliary function with respect to the transform
${\bm W}_r$ , and then maximising it with respect to the transformed mean
yields the following update
\begin{eqnarray}
{{\bm{w}}}_{ri} = {\bf k}_r^{(i)}{\bf G}_r^{(i)-1} \label{eq:mllrmeansol}
\end{eqnarray}

The above expressions assume that each base regression class $r$ has a
separate transform. If regression class trees are used then the shared
transform parameters may be simply estimated by combining the
statistics of the base regression classes.  The regression class tree
is used to generate the classes dynamically, so it is not known
a-priori which regression classes will be used to estimate the
transform. This does not present a problem, since $\bm{G}^{(i)}$ and
$\bm{k}^{(i)}$ for the chosen regression class may be obtained from its
child classes (as defined by the tree). If the parent node $R$ has
children $\left\{R_1,\dots,R_C\right\}$ then
\[
        {\bf k}^{(i)} = \sum_{c=1}^{C} {\bf k}^{(i)}_{R_c}
\]
and
\[
        {\bf G}^{(i)} = \sum_{c=1}^{C} {\bf G}^{(i)}_{R_c}
\]
The same approach of combining statistics from multiple children can
be applied to all the estimation formulae in this section.

\mysubsect{Variance Transformation Matrix ({\tt MLLRVAR}, {\tt MLLRCOV})}{vtransest}

Estimation of the first variance transformation matrices is only available
for diagonal covariance Gaussian systems in the current implementation,
though full transforms can in theory be estimated. The Gaussian covariance is
transformed using\footnote{In the current implementation of the code this
form of transform can only be estimated in addition to the {\tt MLLRMEAN} 
transform},
\[
        \hat{\bm{\mu}}_{m_r} = \bm{\mu}_{m_r}, \:\:\:\: 
        \hat{\bm{\Sigma}}_{r} = \bm{B}_{m_r}^T\bm{H}_r\bm{B}_{m_r}
\]
where $\bm{H}_m$ is the linear transformation to be estimated and
$\bm{B}_m$ is the inverse of the Choleski factor of $\bm{\Sigma}_{m_r}^{-1}$,
so
\[ 
        \bm{\Sigma}_{m_r}^{-1} = \bm{C}_{m_r}\bm{C}_{m_r}^T
\]
and
\[
        \bm{B}_{m_r} = \bm{C}_{m_r}^{-1}
\]
After rewriting the auxiliary function, the transform matrix $\bm{H}_m$
is estimated from,
\[
        \bm{H}_r = \frac{ \sum_{m_r=1}^{M_r}\bm{C}_{m_r}^T 
                          \left[
                            \like(\bm{o}(t) - \hat{\bm{\mu}}_{m_r})
                                   (\bm{o}(t) - \hat{\bm{\mu}}_{m_r})^T
                          \right]
                          \bm{C}_{m_r}
                        }
                        { \like }
\]
Here, $\bm{H}_r$ is forced to be a diagonal transformation by setting
the off-diagonal terms to zero, which ensures that
$\hat{\bm{\Sigma}}_{m_r}$ is also diagonal.

The alternative form of variance adaptation us supported for full,
block and diagonal transforms. Substituting the for expressions for
variance adaptation
\begin{eqnarray}
\hat{\bm{\mu}}_{m_r} = \bm{\mu}_{m_r}, \:\:\:\: 
\hat{\bm{\Sigma}}_{m_r} = {\bm H}_r{\bm{\Sigma}}_{m_r}{\bm H}_r^T
\end{eqnarray}
into the auxiliary function, and using the fact that the covariance
matrices are diagonal yields
\begin{eqnarray}
{\cal Q}({\cal M},{\hat{\cal M}}) = K + 
\frac{1}{2}
\sum_{r=1}^R
\beta_r\log({\bf c}_{ri}{\bf a}_{ri}^T)-
\sum_{j=1}^d{
\left({\bf a}_{rj}{\bf G}^{(j)}_r{\bf a}^T_{rj}
\right)} \nonumber
\end{eqnarray}
where 
\begin{eqnarray}
\beta_r &=& \sum_{m_r=1}^{M_r}\sum_{t=1}^T\like\\
{\bf A}_r &=& {\bf H}_r^{-1}
\end{eqnarray}
${\bf a}_{ri}$ is $i^{th}$ row of ${\bf
A}_r$, the $1\times n$ row vector ${\bf c}_{ri}$ is the vector of
cofactors of ${\bf A}_r$, $c_{rij}={\mbox{cof}}({\bf A}_{rij})$,
and  ${\bf G}^{(i)}_r$ is defined as
\begin{eqnarray}
{\bf G}^{(i)}_r=\sum_{m_r=1}^{M_r}
\frac{1}{\sigma_{m_ri}^{2}}
\sum_{t=1}^T\like(\bm{o}(t)-\hat{\bm{\mu}}_{m_r})
(\bm{o}(t)-\hat{\bm{\mu}}_{m_r})^T
\label{eq:gi_mllr2}
\end{eqnarray}
Differentiating the auxiliary function with respect to the transform
${\bm A}_r$ , and then maximising it with respect to the transformed mean
yields the following update
\begin{eqnarray}
{\bf a}_{ri} ={\bf c}_{ri}{\bf G}^{(i)-1}_r
\sqrt{\left(\frac{\beta_r}{{\bf
c}_{ri}{\bf G}_r^{(i)-1}{\bf c}^T_{ri}}\right)}
\end{eqnarray}
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). For the 
diagonal transform case it is of course non-iterative and simplifies to
the same form as the {\tt MLLRVAR} transform.

\mysubsect{Constrained MLLR Transformation Matrix ({\tt CMLLR})}{cmllrest}

Substituting the for expressions for CMLLR adaptation where\footnote{For
efficiency this transformation is implemented as
\begin{eqnarray}
\hat{\bm o}_r(t) = \bm{A}_r\bm{o}(t) + \bm{b}_r = \bm{W}_r\bm{\zeta}(t)
\end{eqnarray}
}
\begin{eqnarray}
\hat{\bm{\mu}}_{m_r} = \bm{H}_r\bm{\mu}_{m_r} + \tilde{\bm{b}}_r, \:\:\:\: 
\hat{\bm{\Sigma}}_{m_r} = {\bm H}_r{\bm{\Sigma}}_{m_r}{\bm H}_r^T
\end{eqnarray}
into the auxiliary function, and using the fact that the covariance
matrices are diagonal yields
\begin{eqnarray}
{\cal Q}({\cal M},{\hat{\cal M}}) = K + 
\frac{1}{2}
\sum_{r=1}^R\left[
\beta\log({\bf p}_{ri}\bm{w}_{ri}^T)-
\sum_{j=1}^d{
\left(\bm{w}_{rj}{\bf G}^{(j)}_r\bm{w}^T_{rj} - 2\bm{w}_{rj}{\bf k}^{(j)}_r
\right)}\right] \nonumber
\end{eqnarray}
where 
\begin{eqnarray}
\bm{W}_r = \left[\begin{array}{c c}
-\bm{A}_r\tilde{\bm{b}}_r & \bm{H}_r^{-1} \end{array}
\right] =  \left[\begin{array}{c c}
\bm{b} & \bm{A} \end{array}
\right]
\end{eqnarray}
$\bm{w}_{ri}$ is $i^{th}$ row of $\bm{W}_r$, 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
\begin{eqnarray}
{\bf G}^{(i)}_r=\sum_{m_r=1}^{M_r}
\frac{1}{\sigma_{m_ri}^{2}}
\sum_{t=1}^T\like{\bm\zeta}(t){\bm\zeta}^{T}(t)
\label{eq:gi_mllr3}
\end{eqnarray}
and 
\begin{eqnarray}
{\bf k}^{(i)}_r=\sum_{m_r=1}^{M_r}
\frac{\mu_{m_ri}}{\sigma_{m_ri}^{2}}
\sum_{t=1}^T\like{\bm\zeta}^{T}(t)
\label{eq:gi_mllr4}
\end{eqnarray}
Differentiating the auxiliary function with respect to the transform
$\bm{W}_r$ , and then maximising it with respect to the transformed mean
yields the following update
\begin{eqnarray}
\bm{w}_{ri} = \left(\alpha{{\bf p}_{ri}} + {\bf k}^{(i)}_r\right){\bf G}^{(i)-1}_r
\end{eqnarray}
where $\alpha$ satisfies
\begin{eqnarray}
\alpha^2{\bf p}_{ri}{\bf G}^{(i)-1}_r{\bf p}_{ri}^T +
\alpha{\bf p}_{ri}{\bf G}^{(i)-1}_r{\bf k}^{(i)T}_r - \beta=0\label{eq:alpha_quad}
\end{eqnarray}
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).

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