Abstract
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.
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References
Aamodt, A., Plaza, E.: Case-based reasoning: foundational issues, methodological variations, and system approaches. AI Commun. 7(1), 39–59 (1994)
Amari, S.I.: Differential-Geometrical Methods in Statistics, vol. 28. Springer, New York (2012). https://doi.org/10.1007/978-1-4612-5056-2
Boothby, W.M.: An Introduction to Differentiable Manifolds and Riemannian Geometry, vol. 120. Academic Press, New York (1986)
Cencov, N.N.: Statistical Decision Rules and Optimal Inference, vol. 53. American Mathematical Soc., Providence (2000)
Cornuéjols, A., Ales-Bianchetti, J.: Analogy and induction: which (missing) link? In: Workshop “Advances in Analogy Research: Integration of Theory and Data from Cognitive, Computational and Neural Sciences”. Sofia, Bulgaria (1998)
Falkenhainer, B., Forbus, K.D., Gentner, D.: The structure-mapping engine: algorithm and examples. Artif. Intell. 41(1), 1–63 (1989)
Fisher, R.A.: Theory of statistical estimation. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 22, pp. 700–725. Cambridge University Press, Cambridge (1925)
Goswami, U.: Analogical Reasoning in Children. Psychology Press, Hove (2013)
Han, M., Park, F.C.: DTI segmentation and fiber tracking using metrics on multivariate normal distributions. J. Math. Imaging Vis. 49(2), 317–334 (2014)
Hofstadter, D., Mitchell, M.: The copycat project: a model of mental fluidity and analogy-making. In: Fluid Concepts and Creative Analogies, pp. 205–267. Basic Books Inc., New York (1995)
Hwang, S.J., Grauman, K., Sha, F.: Analogy-preserving semantic embedding for visual object categorization. In: Dasgupta, S., Mcallester, D. (eds.) Proceedings of the 30th International Conference on Machine Learning (ICML 2013), vol. 28, pp. 639–647. JMLR Workshop and Conference Proceedings, May 2013. http://jmlr.org/proceedings/papers/v28/juhwang13.pdf
Lepage, Y.: Solving analogies on words: an algorithm. In: Proceedings of the 17th international conference on Computational linguistics, vol. 1, pp. 728–734. Association for Computational Linguistics (1998)
Miclet, L., Bayoudh, S., Delhay, A.: Analogical dissimilarity: definition, algorithms and two experiments in machine learning. J. Artif. Intell. Res. 32, 793–824 (2008). http://dblp.uni-trier.de/db/journals/jair/jair32.html#MicletBD08
Mikolov, T., Chen, K., Corrado, G., Dean, J.: Efficient estimation of word representations in vector space. CoRR abs/1301.3781 (2013). http://dblp.uni-trier.de/db/journals/corr/corr1301.html#abs-1301-3781
Mikolov, T., Yih, W.T., Zweig, G.: Linguistic regularities in continuous space word representations. In: HLT-NAACL, pp. 746–751 (2013)
Ollivier, Y.: A visual introduction to Riemannian curvatures and some discrete generalizations. In: Analysis and Geometry of Metric Measure Spaces: Lecture Notes of the 50th Séminaire de Mathématiques Supérieures (SMS), Montréal, pp. 197–219 (2011)
Rumelhart, D.E., Abrahamson, A.A.: A model for analogical reasoning. Cognit. Psychol. 5(1), 1–28 (1973). http://www.sciencedirect.com/science/article/pii/0010028573900236
Skovgaard, L.T.: A Riemannian geometry of the multivariate normal model. Scand. J. Stat. 11, 211–223 (1984)
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This research is supported by the program Futur & Ruptures (Institut Mines Télécom).
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Murena, PA., Cornuéjols, A., Dessalles, JL. (2018). Opening the Parallelogram: Considerations on Non-Euclidean Analogies. In: Cox, M., Funk, P., Begum, S. (eds) Case-Based Reasoning Research and Development. ICCBR 2018. Lecture Notes in Computer Science(), vol 11156. Springer, Cham. https://doi.org/10.1007/978-3-030-01081-2_39
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