A Comprehensive Implementation of Conceptual Spaces Lucas - PowerPoint PPT Presentation

A Comprehensive Implementation of Conceptual Spaces Lucas Bechberger and Kai-Uwe Kühnberger https://www.lucas-bechberger.de

The Different Layers of Representation ∀ x :apple ( x ) ⇒ red ( x ) Symbolic Layer Formal Logics Geometric ? Conceptual Layer Representation Sensor / Feature Subsymbolic Layer [0.42; -1.337] Values A Comprehensive Implementation of Conceptual Spaces / Lucas Bechberger 2

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 A Comprehensive Implementation of Conceptual Spaces / Lucas Bechberger 3

Betweenness  B(x,y,z) :↔ d(x,y) + d(y,z) = d(x,z)  Convex region C: ∀ x,z ∈ C : ∀ y : B ( x,y,z ) ⇒ y ∈ C  Star-shaped region S: ∃ p ∈ S : ∀ z ∈ S : ∀ y : B ( p,y,z ) ⇒ y ∈ S https://en.wikipedia.org/wiki/Taxicab_geometry#/ media/File:Manhattan_distance.svg A Comprehensive Implementation of Conceptual Spaces / Lucas Bechberger 4

Convexity and Manhattan distance adult height child age A Comprehensive Implementation of Conceptual Spaces / Lucas Bechberger 5

Formalizing Star-Shaped Concepts A Comprehensive Implementation of Conceptual Spaces / Lucas Bechberger 6

Formalizing Star-Shaped Concepts ~ S = S 1.0 ~ S 0.5 ~ S 0.25 A Comprehensive Implementation of Conceptual Spaces / Lucas Bechberger 7

Operations on concepts x v ~ S 1 ~ S 2 ~ S 3 A Comprehensive Implementation of Conceptual Spaces / Lucas Bechberger 8

DEMO TIME! A Comprehensive Implementation of Conceptual Spaces / Lucas Bechberger 9

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 A Comprehensive Implementation of Conceptual Spaces / Lucas Bechberger 10

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