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A Comprehensive Implementation of Conceptual Spaces / Lucas Bechberger 2
A Comprehensive Implementation of Conceptual Spaces Lucas - - PowerPoint PPT Presentation
A Comprehensive Implementation of Conceptual Spaces Lucas - - PowerPoint PPT Presentation
A Comprehensive Implementation of Conceptual Spaces Lucas Bechberger and Kai-Uwe Khnberger https://www.lucas-bechberger.de The Different Layers of Representation x :apple ( x ) red ( x ) Symbolic Layer Formal Logics Geometric ?
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A Comprehensive Implementation of Conceptual Spaces / Lucas Bechberger 3
Conceptual Spaces [Gärdenfors2000]
- Quality dimensions
- Interpretable ways of judging the similarity of two instances
- E.g., temperature, weight, brightness, pitch
- Domain
- Set of dimensions that inherently belong together
- Color: hue, saturation, and brightness
- Distance in this space is inversely related to similarity
- Within a domain: Euclidean distance
- Between domains: Manhattan distance
- Concepts
- Region + correlation information + salience weights
[Gärdenfors2000] Gärdenfors, P. Conceptual Spaces: The Geometry of Thought. MIT press, 2000
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A Comprehensive Implementation of Conceptual Spaces / Lucas Bechberger 4
Betweenness
- B(x,y,z) :↔ d(x,y) + d(y,z) = d(x,z)
- Convex region C:
- Star-shaped region S:
https://en.wikipedia.org/wiki/Taxicab_geometry#/ media/File:Manhattan_distance.svg
∀ x,z∈C :∀ y: B ( x,y,z)⇒ y∈C ∃ p∈S: ∀ z∈S: ∀ y :B ( p,y,z)⇒ y ∈S
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A Comprehensive Implementation of Conceptual Spaces / Lucas Bechberger 5
Convexity and Manhattan distance
height age adult child
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A Comprehensive Implementation of Conceptual Spaces / Lucas Bechberger 6
Formalizing Star-Shaped Concepts
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Formalizing Star-Shaped Concepts
S = S1.0 ~ S0.5 S0.25 ~ ~
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A Comprehensive Implementation of Conceptual Spaces / Lucas Bechberger 8
Operations on concepts
S1 ~ S2 ~ x v
S3 ~
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A Comprehensive Implementation of Conceptual Spaces / Lucas Bechberger 9
DEMO TIME!
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A Comprehensive Implementation of Conceptual Spaces / Lucas Bechberger 10
Research Contributions
- We can encode correlations in a geometric way
- Most formalizations of conceptual spaces ignore cross-domain
correlations
- Publicly available implementation
- https://github.com/lbechberger/ConceptualSpaces
- Comes with source code → easily modifiable
- Comprehensive set of supported operations
- Visualization tool
- Starting point for future research
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