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<Paper uid="H92-1093">
  <Title>Weight Estimation for N-Best Rescoring*</Title>
  <Section position="3" start_page="0" end_page="0" type="metho">
    <SectionTitle>
2. WEIGHT ESTIMATION
</SectionTitle>
    <Paragraph position="0"> In the initial work \[1\], the weights used in the linear score combination were chosen to minimize the generalized mean of the rank of the correct hypothesis using an iterative search algorithm based on Powell's method \[2\].</Paragraph>
    <Paragraph position="1"> Further experience using this technique suggested that the result was very sensitive to the large number of local minima in the optimization criterion.</Paragraph>
    <Paragraph position="2"> Several steps have been taken to address this issue. The optimization criterion now minimizes the average word error in the top ranking hypothesis. The use of this criterion results in a &amp;quot;smoother&amp;quot; weight space, i.e., having fewer local minima. Also addressing the problem of local minima, we examine a large number of points in the weight space on a lattice spanning the range of probable weights. Powell's method may be used with points on the grid as the initial estimate of weights to find the best performance, or the points on a fine grid may be evaluated directly.</Paragraph>
    <Paragraph position="3"> The error function is piece-wise constant over the weight *This research was jointly funded by NSF and DARPA under NSF grant number IRI-8902124.</Paragraph>
    <Paragraph position="4"> space. A particular ranking of the hypotheses corresponds to a region (cell) defined by a set of inequalities that describe a polytope. In the hope of obtaining a more robust estimate, we measure the amount of slack for the different coefficients along the coordinate axes such that the weight remains within the cell as well as determine the &amp;quot;center&amp;quot; of the cell. The product of the slacks in the different coordinate directions at the &amp;quot;center&amp;quot; is an approximate indicator of the &amp;quot;volume&amp;quot; of the cell. If more than one cell gives the same performance, we choose the one with the largest &amp;quot;volume&amp;quot;. Weights which correspond to the &amp;quot;center&amp;quot; of this cell are used for combining scores in the test set.</Paragraph>
  </Section>
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