ABSURDIST II: A Graph Matching Algorithm and its Application to Conceptual System Translation

1
ABSURDIST II A Graph Matching Algorithm and its
Application to Conceptual System Translation

• Ying Feng
• Robert Goldstone
• Vladimir Menkov
• Computer Science Department
• Psychology Department
• Indiana University

2
How concepts get their meanings

• Conceptual web
• A concepts meaning comes from its connections to
  other concepts in the same conceptual system
• External grounding
• A concepts meaning comes from its connection to
  the external world

3
Conceptual Web

• Philosophy
• Conceptual role semantics
• Conceptual incommensurability
• Psychology
• Isolated and interrelated concepts
• Latent semantic analysis
• Computer Science
• Semantic networks
• Intrinsic meaning in large databases

4
Externally Grounded Concepts

• Philosophy
• The symbol grounding problem
• Psychology
• Perceptual symbol systems
• Computer science
• Embodied cognition

5
Translation Across Conceptual Systems

• How can we determine that two people share a
  matching concept of something (such as Mushroom)?
• The publicity of concepts we want to say that
  two people both have a concept of Mushroom even
  though they know different things (Fodor, 1998)
• Cross-person translation as a challenge to
  conceptual web accounts of meaning (Fodor
  Lepore, 1992)
• If a concepts meaning depends on its role in its
  system, and if two people have different systems,
  then they cant have the same meaning

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Problem Definition

• Given Conceptual Systems A and B
• Each has a set of concepts
• Both have the same set of relation types
• Match the concepts in A with those in B based on
• Internal relations between concepts in each
  system
• Partially known (external) correspondence between
  the two systems

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Graph Representation

• Represent conceptual systems as graphs
• Concepts --gt nodes in graph
• Relations --gt links in graph
• Featured graph
• Directed vs undirected
• Labeled vs unlabeled
• Weighted vs unweighted

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Example of Conceptual Systems
moose
seal
reindeer
0.4
0.6
0.8
A sample of Manitoba wildlife
0.9
0.1
polar bear
wolf
Coexists (weighted)
Hunts (unweighted)
élan
phoque
renne
0.6
0.3
0.8
0.9
lours polaire
loup
Un sample de fauna Québécoise
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Graph Matching

• Inputs
• Graphs representing A and B
• External correspondence matrix E
• Output
• Correspondence matrix C (m x n)
• 0 C(Aq, Bx) 1 indicating the correspondence
  between concept Aq and Bx
• Principle of Alignment
• Aligning nodes so that the relations between each
  pair of nodes in one graph are similar to those
  between the aligned nodes in the other graph

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Measuring Similarity of Relation Bundles

• Relation bundle
• All relations between a given pair of nodes
  represented as an M-dimensional vector
• Measuring similarity between Relation Bundles
• Sim(Aq,ArBx,By) 1 - Diff(Aq,ArBx,By)
• Diff(Aq,ArBx,By) ?wi(Aq,Ar)-wi(Bx,By) / M
• 0 Sim 1

Aq
Bx
0.2
1.0
0.6
0.5
Ar
By
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Correspondence Matrix

• C

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Correspondence Matrix

• C

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ABSURDIST II An Optimization Algorithm

• Match quality of a permutation global edge
  similarity measure to maximize
• GlobalEdgeSim(P)
• ß?q,rSim(Aq,ArBP(q),BP(r))
  a?q,rE(Aq,BP(q))

B
A
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Energy Functional

• Generalization to an arbitrary correspondence
  matrix C energy functional to maximize
• K(C) a (E C) ß Excitation(C) - ?
  Inhibition(C)
• External( C ) (E C)
• Reward for C matching E
• Excitation( C )
• ?q,r,x,y Sim(Aq,ArBx,By) C(Aq, Bx) C(Ar, By) /
  (n-1)
• Reward for internal similarity matching
• Inhibition( C )
• (?q,r,xC(Aq,Bx)C(Ar,Bx) ?q,x yC(Aq,Bx) C(Aq,By)
  )/(2(n-1))
• Penalty for non-orthogonality of rows or columns

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Iterative Optimization Process

• Maximize K(C) on cube Q in correspondence matrix
  space
• C0 starting point
• Nt grad K(Ct) net input
• Vt Damp( LNt, Ct ) update
• Ct1 Ct Vt
• Damp() damping function
• Keeps Ct1 in Q
• L learning rate
• Net input has external similarity, excitation,
  and inhibition terms

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Convergence

• Similar to steepest descent, but damping ensures
  convergence to a maximum on cube Q
• Always converges to a maximum (not necessarily
  global)
• Alternatives quadratic programming methods
  (NP-hard)

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Choosing Parameters

• Choose learning rate L
• Stay in cube Q
• Ensure convergence
• Maximize convergence speed

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Choosing Parameters

• Influence of ? /ß on the convergence points of C
• Low ? progress along the principal eigenvector,
  eventualy converge to a matrix of all 1s
• High ? converge to the closest permutation
  matrix
• Heuristic solution choose ? /ß to better balance
  excitation and inhibition
• - Start with a lower ?, adaptively vary
  (generally, increase) it to avoid convergence to
  a matrix of all 1s

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Iteration Costs

• O(n4) terms in the formula for Nt
• Sparse conceptual systems average degree of node
  is d ltlt n
• Exploiting sparsity to compute update in O(n2 d
  2) operations

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Testing ABSURDIST II

• Create conceptual systems for two people
• Create a set of N concepts in Person A
• Define relations radnomly between concepts of
  various labels and with random weights
• Copy these concepts to Person B
• Add noise to B's relations and/or their weights
• Measure ABSURDIST IIs ability to recover true
  alignments
• 200 separate runs
• Initialize each correspondence unit to a certain
  value (0.5)
• Activation passing for a set number of iterations
• Any concepts connected by a unit with more than a
  threshold (0.8) activity are assumed to be aligned

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ABSURDIST II GUI
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Noise Tolerence vs Graph Size

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Noise Tolerence vs Graph Density

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Coverage Noise vs Intensity Noise

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Iteration Steps vs Graph Size

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Iteration Steps vs Graph Density

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External Similarity Seeding

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Potential Applications of ABSURDIST II

• Object recognition
• Within-object relations provide a strong
  constraint for aligning objects
• Analogical reasoning and similarity
• Most models of analogy require highly structured,
  propositional representations ABSURDIST II can be
  applied when similarities are known, but
  structured representations are hard to find
  pictures, words, etc.
• Translating across large databases
• Ontologies, dictionaries,thesauri, etc.

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Conclusions

• Connecting concepts to both the world and each
  other is an attractive option
• These connections are mutually supportive, not
  antagonistic
• Within-system relational information may have
  surprisingly large influences
• Absurdist II improvements on Absurdist I
• Graph representation of arbitrary conceputal
  systems
• Optimization framework
• Better convergence behavior
• Cost reduction from O(n4) down to O(n2 d 2)

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Thank You

• ABSURDIST II website
• http//www.cs.indiana.edu/yingfeng/ABSURDIST/
• Contact Info
• Ying Feng yingfeng_at_cs.indiana.edu
• Robert Goldstone rgoldsto_at_indiana.edu
• Vladimir Menkov vmenkov_at_cs.indiana.edu