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Semantic Spaces

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Abstract

Any natural language can be considered as a tool for producing large databases (consisting of texts, written, or discursive). This tool for its description in turn requires other large databases (dictionaries, grammars etc.). Nowadays, the notion of database is associated with computer processing and computer memory. However, a natural language resides also in human brains and functions in human communication, from interpersonal to intergenerational one. We discuss in this survey/research paper mathematical, in particular geometric, constructions, which help to bridge these two worlds. In particular, in this paper we consider the Vector Space Model of semantics based on frequency matrices, as used in Natural Language Processing. We investigate underlying geometries, formulated in terms of Grassmannians, projective spaces, and flag varieties. We formulate the relation between vector space models and semantic spaces based on semic axes in terms of projectability of subvarieties in Grassmannians and projective spaces. We interpret Latent Semantics as a geometric flow on Grassmannians. We also discuss how to formulate Gärdenfors’ notion of “meeting of minds” in our geometric setting.

O INTERIOR DO EXTERIOR DO INTERIOR

Pascal Mercier

“Nachtzug nach Lissabon”

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References

  1. Alt, H., Godau, M.: Computing the Fréchet distance between two polygonal curves. Int. J. Comput. Geom. Appl. 5, 75–91 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  2. Ammar, G., Martin, C.: The geometry of matrix eigenvalue methods. Acta Appl. Math. 5(3), 239–278 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  3. Arrondo, E.: Projections of Grassmannians of lines and characterization of Veronese varieties. J. Algorithm Geom. 1, 165–192 (2001)

    Google Scholar 

  4. Arrondo, E., Paoletti, R.: Characterization of Veronese varieties via projections in Grassmannians. In: Ciliberto, C., Geramita, A.V., Harbourne, B., Miró-Roig, R.M., Ranestad, K. (eds.) Projective Varieties with Unexpected Properties: A Volume in Memory of Giuseppe Veronese, Walter de Gruyter, (2005)

  5. Bickerton, D., Szathmáry, E.: Biological Foundations and Origin of Syntax. MIT Press, New York (2009)

    Book  Google Scholar 

  6. Budanitsky, A., Hirst, G.: Evaluating WordNet-based measures of semantic distance. Comput. Linguist. 32(1), 13–47 (2006)

    Article  MATH  Google Scholar 

  7. Carlsson, G.: Topology and data. Bull. AMS 46(2), 255–308 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  8. Chiarello, C., Burgess, C., Richards, L., Pollock, A.: Semantic and associative priming in the cerebral hemispheres: some words do, some words don’t \(\ldots \) sometimes, some places. Brain Lang. 38, 7–104 (1990)

    Article  Google Scholar 

  9. Curto, C., Itskov, V.: Cell groups reveal structure of stimulus space. PLoS Comput. Biol. 4(10), 13 (2008)

    Article  MathSciNet  Google Scholar 

  10. Curto, C., Itskov, V., Veliz-Cuba, A., Youngs, N.: The neural ring: an algebraic tool for analysing the intrinsic structure of neural codes. Bull. Math. Biol. 75(9), 1571–1611 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  11. Demailly, J.P.: Vanishing theorems for tensor powers of a positive vector bundle. In: Geometry and analysis on manifolds (Katata/Kyoto, 1987), pp. 86–105, Lecture Notes in Math., Vol. 1339, Springer, New York (1988)

  12. Dhillon, P., Foster, D.P., Ungar, L.H.: Eigenwords: spectral word embeddings. J. Mach. Learn. Res. 16, 3035–3078 (2015)

  13. Dugger, D., Isaksen, D.C.: Hypercovers in topology. Preprint http://www.math.uiuc.edu/K-theory/0528/

  14. Eliasmith, C.: Neurosemantics and categories. In: Handbook of Categorization in Cognitive Science, pp. 1035–1054, Elsevier, Amsterdam (2005)

  15. Gärdenfors, P.: Conceptual Spaces: The Geometry of Thought. Mass. MIT Press, Cambridge (2000)

    Google Scholar 

  16. Gärdenfors, P.: Geometry of Meaning: Semantics Based on Conceptual Spaces. Mass. MIT Press, Cambridge, pp. 343+xii (2014)

  17. Griffiths, P.: On Cartan’s method of Lie groups and moving frames as applied to uniqueness and existence questions in differential geometry. Duke Math. J. 41, 775–814 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  18. Guiraud, P.: The semic matrices of meaning. Soc. Sci. Inf. 7(2), 131–139 (1968)

    Article  MathSciNet  Google Scholar 

  19. Hatcher, A.: Algebraic Topology. CUP, Cambridge (2002)

    MATH  Google Scholar 

  20. Hackl, M.: The syntax-semantics interface. Lingua 130, 66–87 (2013)

    Article  Google Scholar 

  21. Indefrey, P., Levelt, W.J.M.: The spatial and temporal signatures of word production components. Cognition 92, 101–144 (2004)

    Article  Google Scholar 

  22. Lescheniak, J.D., Levelt, W.J.M.: Word frequency effects in speech production: retrieval of syntactic information and of phonological form. Experimental Psychology: Learning. Mem. Cognit. 20, 824–843 (1994)

    Article  Google Scholar 

  23. Lica, L.: The distinction between WHICH and THAT. With Diagrams. http://home.earthlink.net/~llica/wichthat.htm

  24. Lowe, W.: Towards a theory of semantic space. In: Proceedings of the 23rd Conference of the Cognitive Science Society, pp. 576–581 (2001)

  25. Martin, C., Ammar, G.: The geometry of the matrix Riccati equation and associated eigenvalue methods. In: The Riccati equation, pp. 113–126, Comm. Control Eng. Ser. Springer, New York (1991)

  26. Manin, Y.I.: Zipf’s law and L. Levin’s probability distributions. Funct. Anal. Appl. 48(2) (2014). doi:10.107/s10688-014-0052-1. Preprint arXiv:1301.0427

  27. Manin, Y.I.: Neural codes and homotopy types: mathematical models of place field recognition. Moscow Math. J. 15, 1–8 (2015). arXiv:1501.00897

  28. Manin, D.Y.: The right word in the left place: measuring lexical foregrounding in poetry and prose. www.researchgate.net

  29. Manning, C.D., Schuetze, H.: Foundations of Statistical Natural Language Processing. MIT Press, New York (1999)

    Google Scholar 

  30. Mel’čuk, I.: Language: from meaning to text. In: Beck, D. (ed.), Moscow & Boston (2016)

  31. Poeppel, D.: Language: specifying the site of modality-independent meaning. Curr. Biol. 16(21), R930–R932 (2006)

    Article  Google Scholar 

  32. Port, A., Gheorghita, I., Guth, D., Clark, J.M, Liang, C., Dasu, S., Marcolli, M.: Persistent topology of syntax. arXiv:1507.05134

  33. Postnikov, A.: Total positivity, Grassmannians and networks. Preprint arXiv:math/0609764 [math.CO]

  34. Schütze, H., Pedersen, J.: A vector model for syntagmatic and paradigmatic relatedness. In: Making sense of words, pp. 104–113, Oxford (1993)

  35. Segal, G.: Classifying spaces and spectral sequences. Inst. Hautes Etudes Sci. Publ. Math. 34, 105–112 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  36. Turney, P.D., Pantel, P.: From frequency to meaning: vector space models of semantics. J. Artif. Intell. Res. 37, 141–188 (2010)

    MathSciNet  MATH  Google Scholar 

  37. van Valin Jr., R.D.: Exploring the Syntax-Semantics Interface. Cambridge University Press, Cambridge (2005)

    Book  MATH  Google Scholar 

  38. Warglien, M., Gärdenfors, P.: Semantics, conceptual spaces and the meeting of minds. Synthese 190(12), 2165–2193 (2013)

    Article  Google Scholar 

  39. Wittek, P., Darányi, S.: Spectral composition of semantic spaces. In: Song, D., Melucci, M., Frommholz, I., Zhang, P., Wang, L., Arafat, S. (eds.) Quantum Interaction, Lecture Notes Computer Science, vol. 7052, pp. 60–70. Springer, Heidelberg (2011)

  40. Youngs, N.E.: The neural ring: using algebraic geometry to analyse neural rings, p. 108. arXiv:1409.2544 [q-bio.NC]

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Authors and Affiliations

  1. Max-Planck-Institut für Mathematik, Bonn, Germany

    Yuri I. Manin

  2. California Institute of Technology, Pasadena, CA, USA

    Matilde Marcolli

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  1. Yuri I. Manin
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  2. Matilde Marcolli
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Correspondence to Matilde Marcolli.

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Manin, Y.I., Marcolli, M. Semantic Spaces. Math.Comput.Sci. 10, 459–477 (2016). https://doi.org/10.1007/s11786-016-0278-9

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