discmods.tex

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vector.  If the configuration variable \texttt{SAVEASVQ} is set true, then
the output routines in \htool{HParm} will discard the original vectors
and just save the VQ indices in a \texttt{DISCRETE} file. 
Alternatively, \HTK\ will regard any speech vector with \texttt{\_V} set
as being compatible with discrete HMMs.  Thus, it is not necessary
to explicitly create a database of discrete training files if
a set of continuous speech vector parameter files already exists.
Fig.~\href{f:vqtohmm} illustrates this process.
}\index{saveasvq@\texttt{SAVEASVQ}}
\index{targetkind@\texttt{TARGETKIND}}

Once the training data has been configured for discrete HMMs, the 
rest of the training process is similar to that previously described.
The normal sequence is to build a set of monophone models and then
clone them to make triphones.  As in continuous density systems, 
state tying can be used to improve the
robustness of the parameter estimates.  However, in the case of discrete HMMs,
alternative methods based on interpolation are possible.  These are discussed
in section~\ref{s:psmooth}.

\mysect{Tied Mixture Systems}{tiedmix}

\index{tied-mixtures}
Discrete systems have the advantage of low run-time computation.  However,
vector quantisation reduces accuracy and this can lead to poor performance.
As a intermediate between discrete and continuous, a fully tied-mixture
system can be used.
Tied-mixtures are conceptually just another example of the general parameter tying
mechanism built-in to \HTK.  However, to use them effectively in
speech recognition systems a number of storage and computational 
optimisations must be made.  Hence, they are given special treatment in \HTK.

When specific mixtures are tied as in 
\begin{verbatim}
     TI "mix" {*.state[2].mix[1]} 
\end{verbatim}
then a Gaussian mixture component is shared across all of the owners
of the tie.  In this example, all models will share the same Gaussian
for the first mixture component of state 2.  However, if the mixture
component index is missing, then all of the mixture components participating in
the tie are {\it joined} rather than tied. More specifically, the commands
\begin{verbatim}
     JO 128 2.0
     TI "mix" {*.state[2-4].mix} 
\end{verbatim}
has the following effect.  All of the mixture components in states 2 to 4 of
all models are collected into a pool.  If the number of components
in the pool exceeds 128, as set by the preceding join command 
\texttt{JO}\index{jo@\texttt{JO} command}, then
components with the smallest weights are removed until the pool size is exactly
128.  Similarly, if the size of the initial pool is less than 128, then mixture
components are split using the same algorithm as for the Mix-Up \texttt{MU}
command.\index{mixture tying}   All states then share all of the 
mixture components in this pool.  The new mixture weights are chosen to be proportional
to the log probability of the corresponding new mixture component mean with
respect to the original distribution for that state.  The log is used here
to give a wider spread of mixture weights.  All mixture weights are floored
to the value of the second argument of the \texttt{JO} command times 
\texttt{MINMIX}\index{minmix@\texttt{MINMIX}}.

The net effect of the above two commands is to create a set of \texttt{tied-mixture}
HMMs\footnote{Also called {\it semi-continuous} HMMs in the the literature.}
where the same set of mixture components is shared across all states of
all models.  However, the type of the HMM set so created will still be
\texttt{SHARED} and the internal representation will be the same as for
any other set of parameter tyings.   To obtain the optimised representation 
of the tied-mixture weights
described in section~\ref{s:tmix}, the following \htool{HHEd}
\texttt{HK}\index{hk@\texttt{HK} command} command must be issued
\begin{verbatim}
     HK TIEDHS
\end{verbatim}
This will convert the internal representation to the special tied-mixture
form in which all of the tied mixtures are stored in a global table and
referenced implicitly instead
of being referenced explicitly using pointers.

Tied-mixture HMMs work best if the information relating to different sources
such as delta coefficients and energy are separated into distinct data streams.
This can be done by setting up multiple data stream HMMs from the outset.  
However, it is simpler to use the 
\texttt{SS}\index{ss@\texttt{SS} command}
command in \htool{HHEd} to split the data streams of the currently loaded HMM set.
Thus, for example, the command
\begin{verbatim}
     SS 4 
\end{verbatim}
would convert the currently loaded HMMs to use four separate data streams
rather than one.  When used in the construction of tied-mixture HMMs
this is analogous to the use of multiple codebooks in discrete density 
HMMs.

The procedure for building a set of tied-mixture HMMs may be summarised
as follows\index{tied-mixtures!build procedure}
\begin{enumerate}
\item Choose a {\it codebook} size for each data stream and then 
   decide how many Gaussian components will be needed from an initial set of
    monophones to approximately fill this codebook.
    For example, suppose that there are 48 three state monophones.  If
    codebook sizes of 128 are chosen for streams 1 and 2, and
    a codebook size of 64 is chosen for stream 3 then single Gaussian
    monophones would provide enough mixtures in total to fill the codebooks.
\item Train the initial set of monophones.
\item Use \htool{HHEd} to first split the HMMs into the required number of
    data streams, tie
    each individual stream and then convert the tied-mixture HMM set to 
    have the kind \texttt{TIEDHS}.  
    A typical script to do this for four streams would be
\begin{verbatim}
    SS 4
    JO 256 2.0
    TI st1 {*.state[2-4].stream[1].mix}
    JO 128 2.0
    TI st2 {*.state[2-4].stream[2].mix}
    JO 128 2.0
    TI st3 {*.state[2-4].stream[3].mix}
    JO 64 2.0
    TI st4 {*.state[2-4].stream[4].mix}
    HK TIEDHS
\end{verbatim}
\item Re-estimate the models using \htool{HERest} in the normal way.
\end{enumerate}
Once the set of retrained tied-mixture models has been produced, context
dependent models can be constructed using similar methods to those
outlined previously.

When evaluating probabilities in tied-mixture systems, it is often
sufficient to sum just the most likely mixture components since for any
particular input vector, its probability with respect to many of the Gaussian
components will be very low.\index{pruning!in tied mixtures}
\HTK\ tools recognise \texttt{TIEDHS}
HMM sets as being special in the sense that additional optimisations
are possible. When full tied-mixtures are used, then an additional layer of pruning
is applied.  At each time frame, the log probability of the current observation
is computed for each mixture component.  Then only those components which lie within
a threshold of the most likely component are retained.  This 
pruning is controlled by the \texttt{-c} option in 
\htool{HRest}, \htool{HERest} and \htool{HVite}.

\mysect{Parameter Smoothing}{psmooth}

When large sets of context-dependent triphones are built using
discrete models or
tied-mixture models, under-training\index{under-training} can be a 
severe problem since each 
state has a large number of mixture weight parameters to estimate.
The \HTK\ tool \htool{HSmooth} allows these discrete probabilities or
mixture component weights
to be smoothed with the monophone weights using a technique called 
deleted interpolation\index{deleted interpolation}.  

\htool{HSmooth} is used in combination with \htool{HERest}
working in parallel mode.  The training data is split
into blocks and each block is used separately to re-estimate the
HMMs.  However, since \htool{HERest} is in parallel mode, it outputs a dump
file of accumulators instead of updating the models.  \htool{HSmooth} is then
used in place of the second pass of \htool{HERest}.  It reads in the 
accumulator information from each of the blocks, performs deleted
interpolation smoothing on the accumulator values and then outputs
the re-estimated HMMs in the normal way.

\htool{HSmooth}\index{hsmooth@\htool{HSmooth}} implements a 
conventional deleted interpolation scheme.
However, optimisation of the smoothing weights uses a fast 
binary chop\index{binary chop} scheme 
rather than the more usual Baum-Welch approach.
The algorithm for finding the optimal interpolation weights for a given
state and stream is as follows where the description is given in terms
of tied-mixture weights but the same applies to discrete probabilities.  

Assume that \htool{HERest}
has been set-up to output $N$ separate blocks of accumulators.
Let $w_i^{(n)}$ be the $i$'th
mixture weight based on the accumulator blocks $1$ to $N$ but excluding
block $n$, and let $\bar{w}_i^{(n)}$ be the corresponding context
independent weight.  Let $x_i^{(n)}$ be the $i$'th mixture weight 
count for the deleted block $n$.  The derivative of the log
likelihood of the deleted block, given the probability distribution with
weights $c_i = \lambda w_i + (1 - \lambda) \bar{w}_i$ is given by
\begin{equation}
  D(\lambda) = \sum_{n=1}^N \sum_{i=1}^M x_i^{(n)}
   \left[ \frac{w_i^{(n)} - \bar{w}_i^{(n)}}{
            \lambda w_i^{(n)} + (1 - \lambda ) \bar{w}_i^{(n)}} 
   \right]
\end{equation}
Since the log likelihood is a convex function of $\lambda$, this derivative
allows the optimal value of $\lambda$ to be found by a simple
binary chop algorithm, viz.
\begin{verbatim}
     function FindLambdaOpt:
        if (D(0) <= 0) return 0;
        if (D(1) >= 0) return = 1;
        l=0; r=1;
        for (k=1; k<=maxStep; k++){
           m = (l+r)/2;
           if (D(m) == 0) return m;
           if (D(m) > 0) l=m; else r=m;
        }
        return m;
\end{verbatim}

\htool{HSmooth} is invoked in a similar way to \htool{HERest}.
For example, suppose that the directory \texttt{hmm2} contains a set of
accumulator files output by the first pass of \htool{HERest} running in parallel
mode using as source the HMM definitions listed in \texttt{hlist} and 
stored in \texttt{hmm1/HMMDefs}.  Then the command
\begin{verbatim}
    HSmooth -c 4 -w 2.0 -H hmm1/HMMDefs -M hmm2 hlist hmm2/*.acc
\end{verbatim}
would generate a new smoothed HMM set in \texttt{hmm2}.  Here the \texttt{-w}
option is used to set the minimum mixture component weight in any state to
twice the value of \texttt{MINMIX}\index{minmix@\texttt{MINMIX}}.  
The \texttt{-c} option sets the maximum
number of iterations of the binary chop procedure to be 4.



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