doi: 10.17586/2226-1494-2022-22-3-567-573


Quantum-probabilistic SVD: complex-valued factorization of matrix data

S. Kozhisseri, I. A. Surov


Read the full article  ';
Article in English

For citation:
Kozhisseri S., Surov I.A. Quantum-probabilistic SVD: complex-valued factorization of matrix data. Scientific and Technical Journal of Information Technologies, Mechanics and Optics, 2022, vol. 22, no. 3, pp. 567–573. doi: 10.17586/2226-1494-2022-22-3-567-573


Abstract
The paper reports a method for compressed representation of matrix data on the principles of quantum theory. The method is formalized as complex-valued matrix factorization based on standard singular value decomposition. The developed approach establishes a bridge between standard methods of semantic data analysis and quantum models of cognition and decision. According to the quantum theory, real-valued observable quantities are generated by wavefunctions being complex-valued vectors in multidimensional Hilbert-space. Wavefunctions are defined as superpositions of basis vectors encoding composition of semantic factors. Basis vectors are found by singular value decomposition of the initial data matrix transformed to a real-valued amplitude form. Phase-dependent superposition amplitudes are found to optimize approximation of the source data. The resulting model represents the observed real-valued data as generated from a small number of basis wavefunctions superposed with complex-valued coefficients. The method is tested for random matrices of sizes from 3 × 3 to 12 × 12 and dimensionality of latent Hilbert-space from 2 to 4. The best approximation is achieved by encoding latent factors in normalized complex-valued amplitude vectors interpreted as wavefunctions generating the data. In terms of approximation fitness, the developed method surpasses standard truncated SVD of the same dimensionality. The mean advantage over the considered range of parameters is 22 %. The method permits cognitive interpretation in accord with the existing quantum models of cognition and decision. The method can be integrated in the algorithms of semantic data analysis including natural language processing. In these tasks, the obtained improvement of approximation translates to the increased precision of similarity measures, principal component analysis, advantage in classification, and document ranking methods. Integration with quantum models of cognition and decision is expected to boost methods of artificial intelligence and machine learning improving imitation of natural thinking.

Keywords: quantum probability, cognitive modeling, semantic analysis, wavefunction, matrix decomposition

Acknowledgements. The research was funded by a grant of Russian Science Foundation (project number 20-71-001036).

References
  1. Surov I.A., Alodjants A.P. Decision-making models in quantum cognitive science. ITMO University, 2018, 63 p. (in Russian)
  2. Khrennikov A. Quantum-like modeling of cognition. Frontiers in Physics, 2015, vol. 3, pp. 77. https://doi.org/10.3389/fphy.2015.00077
  3. Khrennikov A. Ubiquitous Quantum Structure: From Psychology to Finance. Springer, 2010, 216 p. https://doi.org/10.1007/978-3-642-05101-2
  4. Busemeyer J.R., Bruza P.D. Quantum Models of Cognition and Decision. Cambridge University Press, 2012, 426 p.
  5. Melucci M. Introduction to Information Retrieval and Quantum Mechanics. Springer, 2015, 232 p. https://doi.org/10.1007/978-3-662-48313-8
  6. Dumais S.T. Latent semantic analysis. Annual Review of Information Science and Technology, 2004, vol. 38, no. 1, pp. 188–230. https://doi.org/10.1002/aris.1440380105
  7. Deerwester S., Dumais S.T., Landauer T.K., Furnas G.W., Harshman R.A. Indexing by latent semantic analysis. Journal of the American Society for Information Science. 1990. V. 41 N 6. P. 391–407. https://doi.org/10.1002/(SICI)1097-4571(199009)41:6<391::AID-ASI1>3.0.CO;2-9
  8. Landauer T.K., Dumais S.T. A solution to Plato’s problem: The latent semantic analysis theory of acquisition, induction, and representation of knowledge. Psychological Review, 1997, vol. 104, no. 2, pp. 211–240. https://doi.org/10.1037/0033-295X.104.2.211
  9. Forsythe G.E., Malcolm M.A., Moler C.B. Computer Methods for Mathematical Computations. Prentice Hall, 1977, 259 p.
  10. Bohm D., Hiley B.J. The Undivided Universe: An Ontological Interpretation of Quantum Theory. Routledge, 1993, 397 p.
  11. Virtanen P., Gommers R., Oliphant T.E. et. al SciPy 1.0: fundamental algorithms for scientific computing in Python. Nature Methods, 2020, vol. 17, no. 3, pp. 261–272. https://doi.org/10.1038/s41592-019-0686-2
  12. Wang B., Li Q., Melucci M., Song D. Semantic Hilbert space for text representation learning. Proc. of the World Wide Web Conference (WWW), 2019, pp. 3293–3299. https://doi.org/10.1145/3308558.3313516
  13. Papagni G., Koeszegi S. A pragmatic approach to the intentional stance semantic, empirical and ethical considerations for the design of artificial agents. Minds and Machines, 2021, vol. 31, no. 4, pp. 505–534. https://doi.org/10.1007/s11023-021-09567-6
  14. González F.A., Caicedo J.C. Quantum latent semantic analysis. Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), 2011, vol. 6931, pp. 52–63. https://doi.org/10.1007/978-3-642-23318-0_7
  15. Khrennikov A. Quantum-like model of cognitive decision making and information processing. BioSystems, 2009, vol. 95, no. 3, pp. 179–187. https://doi.org/10.1016/j.biosystems.2008.10.004
  16. Aerts D., Czachor M. Quantum aspects of semantic analysis and symbolic artificial intelligence. Journal of Physics A: Mathematical and General, 2004, vol. 37, no. 12, pp. 123–132. https://doi.org/10.1088/0305-4470/37/12/L01
  17. Bruza P.D., Coley R.J. Quantum logic of semantic space: An exploratory investigation of context effects in practical reasoning. We Will Show Them: Essays in Honour of Dov Gabbay. Ed. by S. Artemov, H. Barringer, S.A. d’Avila Garcez, L.C. Lamb, J. Woods. College Publications, 2005, pp. 339–361.
  18. Aerts D., Sozzo S., Veloz T. Quantum structure of negation and conjunction in human thought. Frontiers in Psychology, 2015, vol. 6, pp. 1447. https://doi.org/10.3389/fpsyg.2015.01447
  19. Li J., Zhang P., Song D., Hou Y. An adaptive contextual quantum language model. Physica A: Statistical Mechanics and its Applications, 2016, vol. 456, pp. 51–67. https://doi.org/10.1016/j.physa.2016.03.003
  20. Surov I.A., Semenenko E., Platonov A.V., Bessmertny I.A., Galofaro F., Toffano Z., Khrennikov A., Alodjants A.P. Quantum semantics of text perception. Scientific Reports, 2021, vol. 11, no. 1, pp. 4193. https://doi.org/10.1038/s41598-021-83490-9
  21. Shaker A., Bessmertny I., Miroslavskaya L.A., Koroleva J.A. A quantum-like semantic model for text retrieval in Arabic. Scientific and Technical Journal of Information Technologies, Mechanics and Optics, 2021, vol. 21, no. 1, pp. 102–108. (in Russian). https://doi.org/10.17586/2226-1494-2021-21-1-102-108
  22. Platonov A., Bessmertny I., Koroleva J., Miroslavskaya L., Shaker A. Vector representation of words using quantum-like probabilities. Studies in Systems, Decision and Control, 2021, vol. 337, pp. 535–546. https://doi.org/10.1007/978-3-030-65283-8_44
  23. Widdows D., Kitto K., Cohen T. Quantum mathematics in artificial intelligence. Journal of Artificial Intelligence Research, 2021, vol. 72, pp. 1307–1341. https://doi.org/10.1613/jair.1.12702
  24. Hofmann T. Unsupervised learning by probabilistic Latent Semantic Analysis. Machine Learning, 2001, vol. 42, no. 1-2, pp. 177–196. https://doi.org/10.1023/A:1007617005950
  25. Vorontsov K., Potapenko A., Additive regularization of topic models. Machine Learning, 2015, vol. 101, no. 1-3, pp. 303–323. https://doi.org/10.1007/s10994-014-5476-6


Creative Commons License

This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License
Copyright 2001-2022 ©
Scientific and Technical Journal
of Information Technologies, Mechanics and Optics.
All rights reserved.

Яндекс.Метрика