A Thorough Formalization of Conceptual Spaces Lucas Bechberger and - - PowerPoint PPT Presentation

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A Thorough Formalization of Conceptual Spaces Lucas Bechberger and - - PowerPoint PPT Presentation

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A Thorough Formalization of Conceptual Spaces Lucas Bechberger and Kai-Uwe Khnberger The Different Layers of Representation A Thorough Formalization of Conceptual Spaces / Bechberger and Khnberger 2 The Different Layers of Representation

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A Thorough Formalization of Conceptual Spaces

Lucas Bechberger and Kai-Uwe Kühnberger

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A Thorough Formalization of Conceptual Spaces / Bechberger and Kühnberger 2

The Different Layers of Representation

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A Thorough Formalization of Conceptual Spaces / Bechberger and Kühnberger 3

The Different Layers of Representation

Symbolic Layer ∀ x:apple (x ) ⇒red(x ) Formal Logics

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A Thorough Formalization of Conceptual Spaces / Bechberger and Kühnberger 4

The Different Layers of Representation

Symbolic Layer Subsymbolic Layer [0.42; -1.337, ...] ∀ x:apple (x ) ⇒red(x ) Formal Logics Sensor / Feature Values

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A Thorough Formalization of Conceptual Spaces / Bechberger and Kühnberger 5

The Different Layers of Representation

Symbolic Layer Subsymbolic Layer [0.42; -1.337, ...] ∀ x:apple (x ) ⇒red(x ) Formal Logics Sensor / Feature Values

?

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A Thorough Formalization of Conceptual Spaces / Bechberger and Kühnberger 6

The Different Layers of Representation

Symbolic Layer Subsymbolic Layer [0.42; -1.337, ...] ∀ x:apple (x ) ⇒red(x ) Formal Logics Sensor / Feature Values Conceptual Layer Geometric Representation

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A Thorough Formalization of Conceptual Spaces / Bechberger and Kühnberger 7

The Different Layers of Representation

Symbolic Layer Subsymbolic Layer [0.42; -1.337, ...] ∀ x:apple (x ) ⇒red(x ) Formal Logics Sensor / Feature Values Conceptual Layer Geometric Representation

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A Thorough Formalization of Conceptual Spaces / Bechberger and Kühnberger 8

The Different Layers of Representation

Symbolic Layer Subsymbolic Layer [0.42; -1.337, ...] ∀ x:apple (x ) ⇒red(x ) Formal Logics Sensor / Feature Values Conceptual Layer Geometric Representation

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A Thorough Formalization of Conceptual Spaces / Bechberger and Kühnberger 9

Conceptual Spaces [Gärdenfors2000]

[Gärdenfors2000] Gärdenfors, P. Conceptual Spaces: The Geometry of Thought. MIT press, 2000

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A Thorough Formalization of Conceptual Spaces / Bechberger and Kühnberger 10

Conceptual Spaces [Gärdenfors2000]

  • Quality dimensions
  • Interpretable ways of judging the similarity of two instances
  • E.g., temperature, weight, brightness, pitch

[Gärdenfors2000] Gärdenfors, P. Conceptual Spaces: The Geometry of Thought. MIT press, 2000

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A Thorough Formalization of Conceptual Spaces / Bechberger and Kühnberger 11

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

[Gärdenfors2000] Gärdenfors, P. Conceptual Spaces: The Geometry of Thought. MIT press, 2000

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A Thorough Formalization of Conceptual Spaces / Bechberger and Kühnberger 12

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

[Gärdenfors2000] Gärdenfors, P. Conceptual Spaces: The Geometry of Thought. MIT press, 2000

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A Thorough Formalization of Conceptual Spaces / Bechberger and Kühnberger 13

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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Betweenness

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Betweenness

  • B(x,y,z) :↔ d(x,y) + d(y,z) = d(x,z)
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A Thorough Formalization of Conceptual Spaces / Bechberger and Kühnberger 16

Betweenness

  • B(x,y,z) :↔ d(x,y) + d(y,z) = d(x,z)

https://en.wikipedia.org/wiki/Taxicab_geometry#/ media/File:Manhattan_distance.svg

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A Thorough Formalization of Conceptual Spaces / Bechberger and Kühnberger 17

Betweenness

  • B(x,y,z) :↔ d(x,y) + d(y,z) = d(x,z)

https://en.wikipedia.org/wiki/Taxicab_geometry#/ media/File:Manhattan_distance.svg

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A Thorough Formalization of Conceptual Spaces / Bechberger and Kühnberger 18

Betweenness

  • B(x,y,z) :↔ d(x,y) + d(y,z) = d(x,z)

https://en.wikipedia.org/wiki/Taxicab_geometry#/ media/File:Manhattan_distance.svg

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A Thorough Formalization of Conceptual Spaces / Bechberger and Kühnberger 19

Betweenness

  • B(x,y,z) :↔ d(x,y) + d(y,z) = d(x,z)

https://en.wikipedia.org/wiki/Taxicab_geometry#/ media/File:Manhattan_distance.svg

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A Thorough Formalization of Conceptual Spaces / Bechberger and Kühnberger 20

Betweenness

  • B(x,y,z) :↔ d(x,y) + d(y,z) = d(x,z)
  • Convex region C:

https://en.wikipedia.org/wiki/Taxicab_geometry#/ media/File:Manhattan_distance.svg

∀ x,z∈C :∀ y: B ( x,y,z)⇒ y∈C

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A Thorough Formalization of Conceptual Spaces / Bechberger and Kühnberger 21

Betweenness

  • B(x,y,z) :↔ d(x,y) + d(y,z) = d(x,z)
  • Convex region C:
  • Star-shaped region S w.r.t. p:

https://en.wikipedia.org/wiki/Taxicab_geometry#/ media/File:Manhattan_distance.svg

∀ x,z∈C :∀ y: B ( x,y,z)⇒ y∈C ∀ z∈S: ∀ y : B (p,y,z)⇒ y∈S

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Convexity and Manhattan distance

height age

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Convexity and Manhattan distance

height age adult child

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Convexity and Manhattan distance

height age adult child

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Convexity and Manhattan distance

height age adult child

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Convexity and Manhattan distance

height age adult child

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Convexity and Manhattan distance

height age adult child

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Convexity and Manhattan distance

height age adult child

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Formalizing Star-Shaped Concepts

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Formalizing Star-Shaped Concepts

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Formalizing Star-Shaped Concepts

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Formalizing Star-Shaped Concepts

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Formalizing Star-Shaped Concepts

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Formalizing Star-Shaped Concepts

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Formalizing Star-Shaped Concepts

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Formalizing Star-Shaped Concepts

S = S1.0 ~

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Formalizing Star-Shaped Concepts

S = S1.0 ~ S0.5 ~

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Formalizing Star-Shaped Concepts

S = S1.0 ~ S0.5 S0.25 ~ ~

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Intersection of Two Concepts

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Intersection of Two Concepts

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Intersection of Two Concepts

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A Thorough Formalization of Conceptual Spaces / Bechberger and Kühnberger 42

Intersection of Two Concepts

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Intersection of Two Concepts

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Intersection of Two Concepts

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Intersection of Two Concepts

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Unification of Two Concepts

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Unification of Two Concepts

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Unification of Two Concepts

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Unification of Two Concepts

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Unification of Two Concepts

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Unification of Two Concepts

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Unification of Two Concepts

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Unification of Two Concepts

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Projection of a Concept

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Projection of a Concept

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Research Contributions

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Research Contributions

  • We can encode correlations in a geometric way
  • Most formalizations of conceptual spaces ignore cross-domain

correlations

  • [Rickard2006] considers correlations, but not in a geometric way

[Rickard2006] Rickard, J. T. A Concept Geometry for Conceptual Spaces. Fuzzy Optimization and Decision Making, Springer Science + Business Media, 2006, 5, 311-329

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Research Contributions

  • We can encode correlations in a geometric way
  • Most formalizations of conceptual spaces ignore cross-domain

correlations

  • [Rickard2006] considers correlations, but not in a geometric way
  • Easily implementable and computationally efficient
  • Cuboid can be represented by two support points
  • Single constraint: cuboids of a concept must intersect

[Rickard2006] Rickard, J. T. A Concept Geometry for Conceptual Spaces. Fuzzy Optimization and Decision Making, Springer Science + Business Media, 2006, 5, 311-329

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Thank you for your attention!

Questions? Comments? Discussions? https://www.lucas-bechberger.de @LucasBechberger

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Example: The Color Domain

https://en.wikipedia.org/wiki/HSL_and_HSV#/media/File:HSL_color_solid_dblcone_chroma_gray.png

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Intersection & Union (Fuzzy Case)

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Example: Fruit Space

round sweet hue round hue sweet

  • range
  • range

Granny Smith Granny Smith apple apple lemon lemon pear pear banana banana

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Envisioned Architecture