A Graphical Model For Simultaneous Partitioning And Labeling

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A Graphical Model For Simultaneous Partitioning
And Labeling

• Philip Cowans
• Martin Szummer
• AISTATS, Jan 2005

Cambridge
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Motivation Interpreting Ink
Hand-drawn diagram
Machine interpretation
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Graph Construction
Vertices are grouped into parts.
Vertices, V
Each part is assigned a label
G
Edges, E
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Labeled Partitions

• We assume
• Parts are contiguous.
• The graph is triangulated.
• Were interested in probability distributions
  over labeled partitions conditioned on observed
  data.

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Conditional Random Fields

• CRFs (Lafferty et. al.) provide joint labeling of
  graph vertices.
• Idea define parts to be contiguous regions with
  same label.
• But
• Large number of labels needed.
• Symmetry problems / bias.

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1
2
-1
-1
3
3
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A Better Approach

• Extend the CRF framework to work directly with
  labeled partitions.
• Complexity is improved dont need to deal with
  so many labels.
• No symmetry problem were working directly with
  the representation in which the problem is posed.

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Consistency

• Let G and H µ V.
• Y (G) and Y (H) are consistent if and only if
• For any vertex in G Å H, Y (G) and Y (H) agree on
  its label.
• For any pair of vertices in G Å H, Y (G) and
  Y (H) agree on their part membership.
• Denoted Y (G) v Y (H).

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Projection

• Projection maps labeled partitions onto smaller
  subgraphs.
• If G µ V then, the projection of Y onto G is the
  unique labeled partition of G which is
  consistent with Y.

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Notation
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Potentials
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The Model

• Unary

• Pairwise

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The Model
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Training

• Train by finding MAP weights on example data with
  Gaussian prior (BFGS).
• We require the value and gradient of the log
  posterior

Normalization
Marginalization
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Prediction

• New data is processed by finding the most
  probable labeled partition.
• This is the same as normalization with the
  summation replaced by a maximization.

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Inference

• These operations require summation or
  maximization over all possible labeled
  partitions.
• The number of terms grows super-exponentially
  with the size of G.
• Efficient computation possible using message
  passing as distribution factors.
• Proof based on Shenoy Shafer (1990).

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Factorization

• A distribution factors if it can be written as a
  product of potentials for cliques on the graph
• This is the case for the (un-normalized) model.
• This allows efficient computation using message
  passing.

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Message Passing
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9
7
1
2
3
4
5
6
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Message Passing
2,9
1,7,8
1,2,3,4
Upstream
Message summarizes contribution from upstream
to the sum for a given configuration of the
separator.
Junction tree constructed from cliques on
original graph.
2,3,4,5
4,5,6
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Message Passing
2,9
1,7,8
1,2,3,4
2,3,4,5
x22
4,5,6
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Message Update Rule

• Update messages (for summation) according to
• Marginals found using
• Z can be found explicitly

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Complexity
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Experimental Results

• We tested the algorithm on hand drawn ink
  collected using a Tablet PC.
• The task is to partition the ink fragments into
  perceptual objects, and label them as containers
  or connectors.
• Training data set was 40 diagrams, from 17
  subjects with a total of 2157 fragments.
• 3 random splits (20 training and 20 test
  examples).

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Example 1
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Example 1
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Example 2
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Example 2
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Example 3
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Example 3
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Labeling Results

• Labelling error fraction of fragments labeled
  incorrectly.
• Grouping error fraction of edges locally
  incorrect.

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Conclusions

• We have presented a conditional model definied
  over labeled partitions of an undirected graph.
• Efficient exact inference is possible in our
  model using message passing.
• Labeling and grouping simultaneously can improve
  labeling performance.
• Our model performs well when applied to the task
  of parsing hand-drawn ink diagrams.

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Acknowledgements

• Thanks to
• Thomas Minka, Yuan Qi and Michel Gagnet for
  useful discussion and providing software.
• Hannah Pepper for collecting our ink database.