Sentence Alignment

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Lecture 7: Sentence Alignment
Gale-church.png
Lecture video: web TODO
Youtube
Exercises: Gale & Church algorithm

Sentence alignment is an essential step in building a translation system. Often, we have some parallel data (texts in the source and target language which are translations of each other) but we don't know exactly which sentences correspond to each other. The task here is to find this correspondence (alignment).

Once sentence alignment is available, we can proceed further by finding word or phrase correspondences within the aligned sentences, but that's a topic for another lecture.

Gale & Church algorithm[1] is an algorithm for sentence alignment. It assumes that documents are already aligned on the level of paragraphs. For each paragraph, it finds which sentences correspond to each other.

It is formulated as a dynamic programming algorithm, quite analogous to Levenshtein distance.

Possible Operations

Similarly to string edit distance, a sentence can be:

  • deleted -- a source-side sentence with no corresponding target-side sentence
  • inserted -- a target-side sentence with no corresponding source-side sentence
  • substituted -- a pair of source- and target-side sentences which correspond to each other 1-1 (ideally, the most frequent scenario)

However, Gale & Church define a few more operations:

  • contraction -- two source-side sentences correspond to one target sentence
  • expansion -- one source-side sentence corresponds to two target sentences
  • merge -- two source-side sentences correspond to two target sentences (but there is not 1-1 correspondence)

Distance Function

A distance measure (or a cost function) is required so that we can look for a minimal solution. Gale & Church observe that length differences (measured in characters) between matching sentences tend to be normally distributed. Let c be the average ratio between sentence lengths (for zero mean, c would be 1), s^{2} be the observed variance, and l_{1},l_{2} lengths of the source and target sentence, respectively. Then we define:

\delta =(l_{2}-l_{1}c)/{\sqrt  {l_{1}s^{2}}}

\delta is a zero-mean, unit-variance, normally distributed random variable. We can use it to define our distance measure as the inverse of the conditional probability of a match given a difference \delta . Following the Bayes' rule and dropping the (constant) denominator, we obtain:

P({\text{match}}|\delta )\propto P(\delta |{\text{match}})\cdot P({\text{match}})

We use -\log P({\text{match}}|\delta ) so that lower cost is better and that we can sum the values in the algorithm and still have a probability distribution (instead of multiplying them).

Gale & Church estimate the prior P({\text{match}}) empirically from the data, see Table 5 in the paper.

The likelihood can be formulated as:

P(\delta |{\text{match}})=2(1-P(|\delta |))

Where P(|\delta |) is the cumulative distribution function for a 0-mean, unit variance normal distribution.

Algorithm Formulation

Let us define some notation (identical to the original paper):

d(x_{1},y_{1},0,0) -- the cost of substituting x_{1} with y_{1}
d(x_{1},0,0,0) -- the cost of deleting x_{1}
d(0,y_{1},0,0) -- the cost of inserting y_{1}
d(x_{1},y_{1},x_{2},0) -- the cost of contracting x_{1} and x_{2} to y_{1}
d(x_{1},y_{1},0,y_{2}) -- the cost of expanding x_{1} to y_{1} and y_{2}
d(x_{1},y_{1},x_{2},y_{2}) -- the cost of merging x_{1},x_{2} with y_{1},y_{2}

Then, the algorithm can be defined very simply using the following recursive formula. Let source-side sentences (within a paragraph) be x_{i},i=1\ldots I and let target-side sentences be y_{j},j=1\ldots J:

D(i,j)=\min {\begin{cases}D(i,j-1)+d(0,y_{j},0,0)\\D(i-1,j)+d(x_{i},0,0,0)\\D(i-1,j-1)+d(x_{i},y_{j},0,0)\\D(i-1,j-2)+d(x_{i},y_{j},0,y_{{j-1}})\\D(i-2,j-1)+d(x_{i},y_{j},x_{{i-1}},0)\\D(i-2,j-2)+d(x_{i},y_{j},x_{{i-1}},y_{{j-1}})\end{cases}}

Again, similarly to string edit distance, the minimum total distance can be read off the table cell (I,J) and backtracking can be used to find the actual alignment.

Other Algorithms & Tools

Exercises

References

  1. William Gale, Kenneth Church. A Program for Aligning Sentences in Bilingual Corpora
  2. Alexandr Rosen. In Search of the Best Method for Sentence Alignment in Parallel Texts
  3. D. Varga, L. Németh, P. Halácsy, A. Kornai, V. Trón, V. Nagy. Parallel corpora for medium density languages
  4. Fabienne Braune, Alexander Fraser. Improved unsupervised sentence alignment for symmetrical and asymmetrical parallel corpora
  5. Rico Sennrich, Martin Volk. Iterative, MT-based sentence alignment of parallel texts