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% Title.
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\title{Generating exact lattices in a WFST framework}

\name{ Daniel Povey$^1$, Mirko Hannemann$^{1,2}$,  \\
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  {Gilles Boulianne}$^3$, {Luk\'{a}\v{s} Burget}$^4$, {Arnab Ghoshal}$^5$, {Milos Janda}$^2$, {Stefan Kombrink}$^2$, \\
  {Petr Motl\'{i}\v{c}ek}$^6$, {Yanmin Qian}$^7$, {Ngoc Thang Vu}$^8$, {Korbinian Reidhammer}$^9$, {Karel Vesel\'{y}}$^2$
    \thanks{Thanks here.. remember Sanjeev.}}
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%%% TODO: fix thanks.
%\thanks{Arnab Ghoshal was supported by the European Community's Seventh Framework 
% Programme under grant agreement no. 213850 (SCALE); BUT researchers were partially 
%supported by Czech MPO project No. FR-TI1/034.}}
\address{$^1$ Microsoft Research, Redmond, WA, {\normalsize \tt} \\
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         $^2$ Brno University of Technology, Czech Republic, {\normalsize \tt} \\
         $^3$ CRIM, Montreal, Canada \ \  $^4$ SRI International, Menlo Park, CA, USA \\
         $^5$ University of Edinburgh, U.K. \ \  $^6$  IDIAP, Martigny, Switzerland  \\
         $^6$ Shanghai Jiao Tong University, China \ \ $^7$ Karlsruhe Institute of Technology, Germany \\
         $^9$ Friedrich Alexander University of Erlangen-Nurnberg, Germany}
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\pagestyle{plain} % was set to empty in spconf.sty... this makes page number appear.

We describe a lattice generation method that is exact, i.e. it satisfies all the 
natural properties we would want from a lattice of alternative transcriptions of
an utterance.  This method is also quite efficient; it does not 
introduce substantial overhead above one-best decoding.  Our method is
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most directly applicable when using WFST decoders where the WFST is
expanded down to the HMM-state level.  It outputs
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lattices that include state-level alignments as well as word labels.
The general idea is to create a state-level lattice during decoding, and to do
a special form of determinization that retains only the best-scoring path
for each word sequence.

  Speech Recognition, Lattice Generation


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In Section~\ref{sec:wfst} we give a Weighted Finite State Transducer
(WFST) interpretation of the speech-recognition decoding problem, in order
to introduce notation for the rest of the paper.  In Section~\ref{sec:lattices}
we define the lattice generation problem, and review previous work.
In Section~\ref{sec:overview} we give an overview of our method,
and in Section~\ref{sec:details} we summarize some aspects of a determinization
algorithm that we use in our method.  In Section~\ref{sec:exp} we give
experimental results, and in Section~\ref{sec:conc} we conclude.
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\section{WFSTs and the decoding problem}
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The graph creation process we use in our toolkit, Kaldi~\cite{kaldi_paper},
is very close to the standard recipe described in~\cite{wfst},
where the Weighted Finite State Transducer (WFST) decoding graph is
  \HCLG = \min(\det(H \circ C \circ L \circ G)),
where $\circ$ is WFST composisition (note: view $\HCLG$ as a single symbol).
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For concreteness we will speak of ``costs'' rather
than weights, where a cost is a floating point number that typically represents a negated
log-probability.  A WFST has a set of states with one distinguished
start state\footnote{This is the formulation that corresponds best with the toolkit we use.},
each state has a final-cost (or $\infty$ for non-final states);
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and there is a set of arcs, where each arc has a weight,
weight (just think of this as a cost for now), an input label and an output
label.  In $\HCLG$, the input labels are the identifiers of context-dependent
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HMM states, and the output labels represent words.  For both the input and output
symbols, the special label $\epsilon$ may appear meaning ``no label is present.''

  \caption{Acceptor $U$ describing the acoustic scores of an utterance}

Imagine we have an utterance, of length $T$, and want to ``decode'' it (i.e. find
the most likely word sequence, and the corresponding alignment).  A WFST
interpretation of the decoding problem is as follows.  We construct an
acceptor, or WFSA, as in Fig.~\ref{fig:acceptor} (an acceptor is represented as a
WFST with identical input and output symbols).  It has $T{+}1$ states,
and an arc for each combination of (time, context-dependent HMM state).  The
costs on these arcs are the negated and scaled acoustic log-likelihoods.
Call this acceptor $U$.  Define
   S \equiv U \circ \HCLG,
which is the {\em search graph} for this utterance.  It has approximately $T+1$ times
more states than $\HCLG$ itself.  The decoding problem is equivalent to finding
the best path through $S$.  The input symbol sequence for this best path represents
the state-level alignment, and the output symbol sequence is the corresponding
sentence.  In practice we do not do a full search of $S$, but use beam pruning.
Let $B$ be the searched subset of $S$, containing a subset of the states and arcs
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of $S$ obtained by some heuristic pruning procedure.  
When we do Viterbi decoding with beam-pruning, we are finding the best path through $B$. 
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Since the beam pruning is a part of any practical search procedure and cannot
easily be avoided, we will define the desired outcome of lattice generation in terms
of the visited subset $B$ of the search graph $S$.

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\section{The lattice generation problem, and previous work}

\subsection{Lattices, and the lattice generation problem}
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There is no generally accepted single definition of a lattice.  In~\cite{efficient_general}
and~\cite{sak2010fly}, it is defined as a labeled, weighted, directed acyclic graph
(i.e. a WFSA, with word labels).  In~\cite{ney_word_graph}, time information
is also included.  In the HTK lattice format~\cite{htkbook}, phone-level time alignments 
are also supported (along with separate language model, acoustic and pronunciation-probability 
scores), and in~\cite{saon2005anatomy}, state-level alignments are also produced.
In our work here we will be producing state-level alignments; in fact, the input-symbols
on our graph, which we call {\em transition-ids}, are slightly more fine-grained
than acoustic states and contain sufficient information to reconstruct the phone
sequence.  Hence, our lattices contain at least as much information as any prior work
we cite.

There is, as far as we know, no generally accepted problem statement
for lattice generation, but all the the authors we cited seem to
be concerned with the accuracy of the information in the lattice (e.g. that the
scores and alignments are correct) and the completeness of such information (e.g.
that no high-scoring word-sequences are missing).  The simplest
way to formalize these concerns is to express them in terms of a lattice 
pruning beam $\alpha > 0$ (interpret this as a log likelihood difference).
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  \item The lattice should have a path for every word sequence within $\alpha$ of the best-scoring one.
  \item The scores and alignments in the lattice should be accurate.
  \item The lattice should not contain duplicate paths with the same word sequence.
Actually, this definition is not completely formalized-- we have left
the second condition (accuracy of scores and alignments) a little vague.
Let the lattice be $L$.  The way we would like to state this requirement is:
  \item For every path in $L$, the score and alignment corresponds to 
   the best-scoring path in in $B$ for the corresponding word
   sequence\footnote{Or one of the best-scoring paths, in case of a tie}.
The way we actually have to state the requirement in order to get an efficient procedure is:
  \item For every word-sequence in $B$ within $\alpha$ of the best one, the score and alignment
    for the corresponding path in $L$ is accurate.
  \item All scores and alignments in $L$ correspond to actual paths through $B$ (but not always
   necessarily the best ones).
The issue is that we want to be able to prune $B$ before generating a lattice from it,
but doing so could cause paths not within $\alpha$ of the best one to be lost, so we have
to weaken the condition.  In fact this weakening is of very little consequence, since regardless
of pruning, any word-sequence not within $\alpha$ of the best one could be omitted altogether, 
which is the same as being assigned a cost of $\infty$).

We note that by ``word-sequence'' we mean a sequence of whatever symbols are on the
output of $\HCLG$.  In our experiments these symbols represent words, but not including
silence, which we represent via alternative paths in $L$.

\subsection{Previous lattice generation methods}
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Lattice generation algorithms tend to be closely linked to a particular type of decoder,
but are often justified by the same kinds of ideas.
A common assumption underlying lattice generation methods is the {\em word-pair assumption}
of~\cite{ney_word_graph}.  This is the notion that the time boundary between a pair of words
is not affected by the identity of any earlier words.  In a decoder in which there is
a different copy of the lexical tree for each preceding word, this assumption is automatically
satisfied, and if the word-pair assumption holds, in order to generate an accurate lattice it is
sufficient to store a single Viterbi back-pointer at the word level; the entire set of
such back-pointers contains enough information to generate the lattice.  Authors who have used
this type of lattice generation method~\cite{ney_word_graph,odell_thesis} have generally
not been able to evaluate how correct the word-pair assumption is in practice, but it seems
unlikely to cause problems.  Such methods are not applicable for us anyway, as we
use a WFST based decoder in which each copy of the lexical tree does not have a unique
one-word history.

The lattice generation method described in~\cite{efficient_general} 
is applicable to decoders that use WFSTs~\cite{wfst}
expanded down to the $C$ level (i.e. $CLG$), so the input symbols represent
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context-dependent phones.  In WFST based decoding networks, states normally do
not have a unique one-word history, but the authors of~\cite{efficient_general}
were able to satisfy a similar condition at the phone level.  Their method was
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to store a single Viterbi back-pointer at the phone level; use this to 
create a phone-level latice; prune the resulting
lattice; project it to leave only word labels; and then convert this into a word-level
lattice via epsilon removal and determinization.
Note that the form of pruning referred to here is not the same as beam pruning as
it takes account of both the forward and backward parts of the cost.
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The paper also reported experiments with a different method that did not require any
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phone-pair assumption; these experiments showed that the more
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efficient method that relied on the phone-pair assumption
had almost the same the lattice oracle error rate as the other method.  However, the 
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experiments did not evaluate how much impact the assumption had on the accuracy of the scores, 
and this information could be important in some applications.

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The lattice generation algorithm that was described in~\cite{saon2005anatomy}
is applicable to WFSTs expanded down to the $H$ level (i.e. $\HCLG$),
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so the input symbols represent context-dependent states.  It keeps both score and 
state-level alignment information.  In some sense this algorithm also relies
on the word-pair assumption, but since the copies of the lexical tree in the decoding
graph do not have unique word histories, the resulting algorithm has to be quite
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different.  Viterbi back-pointers at the word level are used, but the algorithm keeps track
of not just a single back-pointer in each state, but the $N$ best back-pointers for the $N$ top-scoring distinct
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preceding.  Therefore, this algorithm has more in common with the sentence N-best
algorithm than with the Viterbi algorithm.  By limiting $N$ to be quite small (e.g. $N{=}5$)
the algorithm was made efficient, but this is at the cost of possibly losing word sequences
that would be within the lattice-generation beam.  In addition, this algorithm was quite
complex to implement efficiently.

\section{Overview of our algorithm}
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\subsection{Version without alignments}

In order to explain our algorithm in the easiest way, we will first explain how it would
be if we did not keep the alignment information, and were storing only a single cost
(i.e. the total acoustic plus language-model cost).  This is just for didactic purposes;
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we have not implemented this simple version.
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In this case our algorithm would be quite similar
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to~\cite{efficient_general}, except at the state level rather than the phone level.
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We actually store forward rather than backward pointers: for each active state on each
frame, we create a forward link record for each active arc out of that state; this points
to the record for the destination state of the arc on the next frame (or on 
the current frame, for $\epsilon$-input
arcs).   As in~\cite{efficient_general}, at the end of the utterance
we prune the resulting graph to discard any paths that are not within the beam $\alpha$ of the best cost. 
Let the pruned graph be $P$, i.e.
  P = \mathrm{prune}(B, \alpha),
where $B$ is the un-pruned state-level lattice.
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We project on the output labels (i.e. we keep only the word labels), then remove $\epsilon$
arcs and determinize.  In fact, we use a determinization algorithm that does $\epsilon$
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removal itself.

As in~\cite{efficient_general}, to save memory we actually do the pruning
periodically rather than waiting for the end of the file (we do it every 25 frames).
Our method is equivalent to their method of linking all currently active states to a ``dummy''
final state and then pruning in the normal way.  However, we implement it in such a way
that the pruning algorithm does not always have to go back to the beginning of the utterance.
For each still-active state, we store the cost difference between the best path including that
state, and the best overall path.  This quantity does not always change between different
iterations of calling the pruning algorithm, and when we detect that these quantities are 
unchanged for a particular frame, the pruning algorithm can stop going backward in time.

After the determinization phase, we prune again using the beam $\alpha$.  This is needed because
the determinization process can introduce a lot of unlikely arcs.  In fact, for particular
utterances the determinization process can cause the lattice to expand enough to
exhaust memory.  To deal with this, we currently just detect when determinization
has produced more than a pre-set maximum number of states, then we prune with a tighter
beam and try again.  In future we may try more sophisticated methods such as a
determinization algorithm that does pruning itself.

This ``simple'' version of the algorithm produces an acyclic, deterministic
WFSA with words as labels.  This is sufficient for applications such as language-model

\subsection{Keeping separate graph and acoustic costs}

A fairly trivial extension of the algorithm described above is to store separately
the acoustic costs and the costs arising from $\HCLG$.  This enables us to do things
like generating output from the lattice with different acoustic scaling factors.
We refer to these two costs as the graph cost and the acoustic cost, since the cost
in $\HCLG$ is not just the language model cost but also contains components
arising from transition probabilities and pronunciation probabilities.  We implement
this by using a semiring that contains two real numbers (one for the graph and one
for the acoustic costs); it keeps track of the two costs separately, but its
$\oplus$ operation returns whichever pair has the lowest sum of costs (graph plus acoustic).

Formally, if each weight is a pair $(a,b)$, then $(a,b) \otimes (c,d) = (a{+}c, b{+}d)$,
and $(a,b) \oplus (c,d)$ is equal to $(a,b)$ if $a{+}b < c{+}d$ or if $a{+}b = c{f}+d$ and $a{-}b < c{-}d$,
and otherwise is equal to $(c,d)$.  This is equivalent to the lexicographic semiring 
of~\cite{roark2011lexicographic}, on the pair $((a{+}b),(a{-}b))$.

\subsection{Keeping state-level alignments}

It is useful for various purposes, e.g. discriminative training and certain kinds of
acoustic rescoring, to keep the state-level alignments in the lattices.  We will now
explain how we can make the alignments ``piggyback'' on top of the computation
defined above, by encoding them in a special semiring.

First, let us define $Q = \inv(P)$, i.e. $Q$ is the inverted, pruned state-level lattice,
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where the input symbols are the words and the output symbols are the p.d.f. labels.
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We want to process $Q$ in such a way that we keep only the best path through it
for each word sequence, and get the corresponding alignment.  This is possible
by defining an appropriate semiring and then doing normal determinization.  Firstly, 
let us ignore the fact that we are keeping track of separate graph and acoustic costs, 
to avoid complicating the present discussion.  

We will define a semiring in which symbol sequences are encoded into the weights.
Let a weight be a pair $(c, s)$, where $c$ is a cost and $s$ is a sequence of symbols.
We define the $\otimes$ operation as $(c, s) \otimes (c', s') = (c+c', (s,s'))$, where
$(s,s')$ is $s$ and $s'$ appended together.  We define the $\oplus$ operation so that
it returns whichever pair has the smallest weight: that is, $(c,s) \oplus (c',s')$ 
equals $(c,s)$ if $c < c'$, and $(c',s')$ if $c > c'$.  If the weights are identical,
we cannot arbitrarily return the first pair because this would not satisfy the semiring
axioms.  In this case, we return the pair with the shorter string part, and if
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the lengths are the same, whichever string appears first in dictionary order. 

Let $E$ be an encoding of the inverted state-level lattice $Q$ 
as described above, with the same number of states and arcs; $E$ is an acceptor, with its symbols
equal to the input symbol (word) on the corresponding arc of $Q$, and
the weights on the arcs of $E$ containing both the the weight and the output symbol (p.d.f.)
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on the corresponding arcs of $Q$.  Let $D = \mathrm{det}(\mathrm{rmeps}(E))$.

In the case we are interested in, determinization
will always succeed because $E$ is acyclic (as long as the original decoding
graph $\HCLG$ has no $\epsilon$-input cycles).  Because $D$ is deterministic
and $\epsilon$-free, it has only one path for each word sequence. 
Determinization preserves equivalence,
and equivalence is defined in such a way that the $\oplus$-sum of the
weights of all the paths through $E$ with a particular word-sequence, must be the same
as the weight of the corresponding path through $D$ with that word-sequence.
It is clear from the definition of $\oplus$ that this path through
$D$ has the cost and alignment of the lowest-cost path through $E$ that has the
same word-sequence on it.  

\subsection{Summary of our algorithm}

We now summarize the lattice generation algorithm.  During decoding, we create a data-structure
corresponding to a full state-level lattice.  That is, for every arc of $\HCLG$ we traverse
on every frame, we create a separate arc in the state-level lattice.  These arcs
contain the acoustic and graph costs separately.  We prune the state-level graph using
a beam $\alpha$; we do this periodically (every 25 frames) but this is equivalent to
doing it just once at the end (similar to~\cite{efficient_general}).  Let the pruned
state-level lattice that we get after pruning it at the end of the utterance be $P$.

Let $Q = \inv(P)$, and let $E$ be an encoded version of $Q$ as described above (with the
state labels as part of the weights).  The final lattice we output is
   L = \mathrm{prune}(\mathrm{det}(\mathrm{rmeps}(E)), \alpha) .
The determinization and epsilon removal are done together by a single algorithm
that we will describe below.

The resulting lattice $L$ is a deterministic, acyclic weighted acceptor with the 
words as the labels, and the graph and acoustic costs and the alignments
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encoded into the weights.  The costs and alignments are not ``synchronized'' 
with the words.
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\section{Details of our determinization algorithm}
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We implemented $\epsilon$ removal and determinization as a single algorithm 
because $\epsilon$-removal using the traditional approach would greatly
increase the size of the state-level lattice (this is mentioned 
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in~\cite{efficient_general}).  Our algorithm uses data-structures
specialized for the particular type of weight we are using.  The issue
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is that the determinization process often has to append a single symbol to
a string of symbols (for the state alignments), and the easiest way to
do this in ``generic'' code would involve copying the whole sequence each time.
Instead we use a data structure that enables this to be done in linear
time (it involves a hash table).  

We will briefly describe another unique aspect of our
algorithm.  Determinization algorithms involve weighted subsets of states,
of the form
   S = \{ (s_1, w_1), (s_2, w_2), \ldots \} .
Let this weighted subset, as it appears in a conventional determinization
algorithm with epsilon removal, be the {\em canonical representation} of a state.
A typical determinization algorithm would maintain a map from this representation
to a state index.  We define a {\em minimial representation} of a state
to be like the canonical representation, but only keeping states that
are either final, or have non-$\epsilon$ arcs out of them.  We maintain
a map from the minimal representation to the state index.  We can
show that this algorithm is still correct (it will tend to give output
with fewer states, i.e. more minimal).  As an optimization for speed, we also define
the {\em initial representation} to be the same type of subset, but prior
to following through the $\epsilon$ arcs, i.e. it only contains the states
that we reached by following non-$\epsilon$ arcs from a previous determinized
state.  We maintain a separate map from the initial representation to
the output state index; think of this as a ``lookaside buffer'' that helps us
avoid the expense of following $\epsilon$ arcs. 
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%% The algorithm for $\epsilon$-removal and determinization is also optimized
%% for our particular case.  We will first explain what a generic $\epsilon$-removal
%% and determinization algorithm would look like.  We will explain this using
%% generic weights.  Let us use the notation
%% \begin{equation}
%%    S = \{ (s_1, w_1), (s_2, w_2), \ldots \}
%% \end{equation}
%% for a weighted subset of states in the original FST.  Each subset $S$ corresponds
%% to a state in the determinized FST, i.e. we maintain a map $m(S) \rightarrow s$
%% where $s$ is a state number in the output (in practice there are tolerances
%% involved in this map when the weights contain floating-point numbers).  We need
%% to normalize these subsets to remove any ``common part'' of the weights.  We
%% need to define an associative and commutative ``common divisor'' operation 
%% on weights (call this $\boxplus$; it can be the same as $\oplus$ for many semirings),
%% that gives us the normalizer for a pair of weights.
%% We must be able to  left-divide each individual weight by the result of this operation, i.e.
%% if $c = a \boxplus b$, then we need to be able to solve $a = c a'$ and $b = c b'$ for $a'$ and $b'$.
%% Here, $a'$ and $b'$ become the ``normalized weights''.  Also $\otimes$-multiplication
%% must be left-distributive over this operation, which ensures the right kind of
%% ``uniqueness'' of the normalization operation.  In the particular case of interest
%% $w$ is a pair $(w', s)$ of a ``base-weight'' and a string $s$, and the 
%% operation $(a,s) \boxplus (b,t)$ returns $(a\oplus b, u)$ where $u$ is the longest
%% common prefix of $s$ and $t$ and $a\oplus b$ essentially returns the minimum cost.

%% For the determinization algorithm, we define
%% $\mathrm{normalizer}(S)$ to be the $\boxplus$-sum of the weights in the subset $S$,
%% and $\mathrm{normalize}(S)$ to be $S$ with each weight left-divided by 
%% $\mathrm{normalizer}(S)$.  We also need to define an operation that ``follows through''
%% $\epsilon$ transitions.  Call this $\mathrm{follow-eps}(S)$ 

%% The key characteristic of our method is that unlike all previously reported 
%% methods that we are aware of (with the sole exception of the inefficient method used 
%% as a baseline in~\cite{efficient_general}), our method is exact even if the word-pair 
%% assumption is invalid.  In addition, we are able to generate a lattice with full
%% state-level alignment information, which would not be possible using the approaches
%% used in~\cite{efficient_general}.  Our algorithm also does not need to store $N$
%% back-pointers like the lattice-generation algorithm previously reported in~\cite{saon2005anatomy}
%% for $H$-level WFSTs (i.e. fully-expanded WFSTs).  

\section{Experimental results}

We report experimental results on the Wall Street Journal database of
read speech.  
Our system is a standard mixture-of-Gaussians system trained on the SI-284
training data; we test on the November 1992 evaluation data.  
For these experiments we generate lattices with the bigram language model
supplied with the WSJ database, and rescore them with the trigram language model.


Conclusions here
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max_arcs=50000 (75000 for lat_beam 10.0)


333 utterances
87 utterances contain OOV
42 utterances contain annotated noise (actually some more, but listed as OOV utterances here)

5641 tokens (some scripts count 5700 with 59 noise annotations)
107 OOV tokens
1.88% OOV token rate

253204 frames (2532 seconds)

19976 words in decoding vocabulary and LM (not counting <s>,</s>,<unk>)

average lattice link density per frame:
have a wide range over files: for example beam=15.0, lat_beam=9.0:
ranging from 1.5 to 167

running on blades, using 64bit version with double float and lattice-simple-decoder

decoding with bigram LM:
decoding WER:      11.51%
ins/del/sub:       118/57/474   
utterances wrong:  227          

experiment: changing lattice beam:

decoding beam:     15.0         15.0       15.0       15.0       15.0         15.0        15.0        15.0        15.0         15.0        15.0
lattice beam:      10.0          9.0        8.0        7.0        6.0          5.0         4.0         3.0         2.0          1.0         0.0
avg. link density: 23.56        16.0       10.82       7.26       4.91         3.38        2.38        1.77        1.4          1.17        1.0
realtime factor:    3.61         3.17       2.59       2.41       2.34         2.98 ?      3.21 ?      1.89        3.12 ?       1.78        --

oracle WER:         2.62%        2.78%      2.85%      3.08%      3.35%        3.86%       4.52%       5.46%       6.98%        8.62%      11.51%
ins/del/sub:       22/0/126     25/0/132   29/0/132   31/1/142   38/5/146     51/11/156   57/15/183   64/20/224   76/32/286    89/40/357   118/57/474
oracle utt wrong:  93           96         96         101        105          116         129         147         168          191         227

rescoring WER(15):  9.59%        9.59%      9.59%      9.59%      9.56%        9.59%       9.61%       9.70%       9.80%       10.21%      11.51%
ins/del/sub:       117/36/388                                    116/36/387   117/35/389  115/35/392  111/38/398  108/39/406   113/42/421  118/57/474
rescore utt wrong: 205                                           204          204         204         202         205          212         227