Opening the Parallelogram: Considerations on Non-Euclidean Analogies

Publikationstyp
Conference Paper
Date Issued
2018-07
Sprache
English
Author(s)
Murena, Pierre Alexandre  
CornuĂ©jols, Antoine  
Dessalles, Jean Louis  
TORE-URI
http://hdl.handle.net/11420/15256
Journal
Lecture notes in computer science  
Volume
11156 LNAI
Start Page
597
End Page
611
Citation
26th International Conference on Case-Based Reasoning (ICCBR 2018)
Contribution to Conference
26th International Conference on Case-Based Reasoning, ICCBR 2018  
Publisher DOI
10.1007/978-3-030-01081-2_39
Scopus ID
2-s2.0-85055717095
Analogical reasoning is a cognitively fundamental way of reasoning by comparing two pairs of elements. Several computational approaches are proposed to efficiently solve analogies: among them, a large number of practical methods rely on either a parallelogram representation of the analogy or, equivalently, a model of proportional analogy. In this paper, we propose to broaden this view by extending the parallelogram representation to differential manifolds, hence spaces where the notion of vectors does not exist. We show that, in this context, some classical properties of analogies do not hold any longer. We illustrate our considerations with two examples: analogies on a sphere and analogies on probability distribution manifold.
Subjects
Analogy
Non-Euclidean geometry