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<Paper uid="P91-1018">
  <Title>regier@cogsci.Berkeley.ED U * TR &amp;quot;Above&amp;quot; Figure 1: Learning to Associate Scenes with Spatial Terms</Title>
  <Section position="4" start_page="0" end_page="139" type="intro">
    <SectionTitle>
2 Generalization and Parameterized
</SectionTitle>
    <Paragraph position="0"/>
    <Section position="1" start_page="0" end_page="138" type="sub_section">
      <SectionTitle>
Regions
2.1 The Problem
</SectionTitle>
      <Paragraph position="0"> The problem of learning whether a particular point lies in a given region of space is a foundational one, with several widely-known &amp;quot;classic&amp;quot; solutions \[Minsky and Papert, 1988; Rumelhart and McClelland, 1986\]. The task at hand is very similar to this problem, since learning when &amp;quot;above&amp;quot; is an appropriate description of the spatial relation between a LM and a point TR really amounts to learning what the extent of the region &amp;quot;above&amp;quot; a LM is.</Paragraph>
      <Paragraph position="1"> However, there is an important difference from the classic problem. We are interested here in learning whether or not a given point (the TR) lies in a region (say &amp;quot;above&amp;quot;, &amp;quot;in&amp;quot;) which is itself located relative to a LM. Thus, the shape, size, and position of the region are dependent on the shape, size, and position of the current LM. For example, the area &amp;quot;above&amp;quot; a small triangle toward the top of the visual field will differ in shape, size,  and position from the area &amp;quot;above&amp;quot; a large circle in the middle of the visual field.</Paragraph>
    </Section>
    <Section position="2" start_page="138" end_page="138" type="sub_section">
      <SectionTitle>
2.2 Parameterized Regions
</SectionTitle>
      <Paragraph position="0"> Part of the solution to this problem lies in the use of parameterized regions. Rather than learn a fixed region of space, the system learns a region which is parameterized by several features of the LM, and is thus dependent on them.</Paragraph>
      <Paragraph position="1"> The LM features used are the location of the center of mass, and the locations of the four corners of the smallest rectangle enclosing the LM (the LM's &amp;quot;bounding-box&amp;quot;). Learning takes place relative to these five &amp;quot;key points&amp;quot;. Consider Figure 2. The figure in (a) shows a region in 2-space learned using the intersection of three halfplanes, as might be done using an ordinary perceptron.</Paragraph>
      <Paragraph position="2"> In (b), we see the same region, but learned relative to the five key points of an LM. This means simply that the lines which define the half-planes have been constrained to pass through the key points of the LM. The method by which this is done is covered in Section 5. Further details can be found in \[Re#eL 1990\].</Paragraph>
      <Paragraph position="3"> The critical point here is that now that this region has been learned relative to the LM key points, it will change position and size when the LM key points change. This is illustrated in (c). Thus, the region is parameterized by the LM key points.</Paragraph>
    </Section>
    <Section position="3" start_page="138" end_page="139" type="sub_section">
      <SectionTitle>
2.3 Combining Representations
</SectionTitle>
      <Paragraph position="0"> While the use of parameterized regions solves much of the problem of generalizability across LMs, it is not sufficient by itself. Two objects could have identical key points, and yet differ in actual shape. Since part of the definition of &amp;quot;above&amp;quot; is that the TR is not in the interior of the LM, and since the shape of the interior of the LM cannot be derived from the key points alone, the key points are an underspecification of the LM for our purposes.</Paragraph>
      <Paragraph position="1"> The complete LM specification includes a bitmap of the interior of the LM, the &amp;quot;LM interior map&amp;quot;. This is simply a bitmap representation of the LM, with those bits set which fall in the interior of the object. As we shall see in greater detail in Section 5, this representation is used together with parameterized regions in learning the perceptual grounding for spatial term semantics. This bitmap representation helps in the case mentioned above, since although the triangle and square will have identical key points, their LM interior maps will differ.</Paragraph>
      <Paragraph position="2"> In particular, since part of the learned &amp;quot;definition&amp;quot; of a point being above a LM should be that it may not be in the interior of the LM, that would account for the difference in shape of the regions located above the square and above the triangle.</Paragraph>
      <Paragraph position="3"> Parameterized regions and the bitmap representation, when used together, provide the system with the ability to generalize across LMs. We shall see examples of this after a presentation of the second major problem to be tackled.</Paragraph>
    </Section>
  </Section>
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