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Quantum Foundations, Probability and Information pp 215–228Cite as

Classification Problem in a Quantum Framework

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Abstract

The aim of this paper is to provide a quantum counterpart of the well-known minimum-distance classifier named Nearest Mean Classifier (NMC). In particular, we refer to the following previous works: (1) in Sergioli et al. Sergioli, G., Santucci, E., Didaci, L., Miszczak, J., Giuntini, R.: A quantum-inspired version of the Nearest Mean Classifier. Soft Computing, 22(3), 691–705 (2018). we have introduced a detailed quantum version of the NMC, named Quantum Nearest Mean Classifier (QNMC), for two-dimensional problems, and we have proposed a generalization to arbitrary dimensions; (2) in Sergioli et al. (Int J Theor Phys 56(12):3880–3888, 2017) the n-dimensional problem was analyzed in detail, and a particular encoding for arbitrary n-feature vectors into density operators has been presented. In this paper, we introduce a new promising encoding of arbitrary n-dimensional patterns into density operators, starting from the two-feature encoding provided in Sergioli et al. Sergioli, G., Santucci, E., Didaci, L., Miszczak, J., Giuntini, R.: A quantum-inspired version of the Nearest Mean Classifier. Soft Computing, 22(3), 691–705 (2018). Further, unlike the NMC, the QNMC shows to be not invariant by rescaling the features of each pattern. This property allows us to introduce a free parameter whose variation provides, in some case, an improvement of the QNMC performance. We show experimental results where i) the NMC and QNMC performances are compared on different datasets and ii) the effects of the non-invariance under uniform rescaling for the QNMC are investigated.

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Notes

  1. 1.

    We remind that, given a function f : X → Y , the argmin (i.e., the argument of the minimum) over some subset S of X is defined as: \(\operatorname *{argmin}_{x \in S\subseteq X } f(x) = \{x|x \in S \wedge \forall y\in S : f(y) \geq f(x)\}\). In this framework, the argmin plays the role of the classifier, i.e., a function that associates to any unlabeled object the correspondent label.

  2. 2.

    We consider the representation of an arbitrary density operator as linear combination of Pauli matrices.

  3. 3.

    http://archive.ics.uci.edu/ml.

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Acknowledgements

This work is supported by the Sardinia Region Project “Time-logical evolution of correlated microscopic system”, LR 7/8/2007 (2015). RAS CRP-55.

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Authors and Affiliations

  1. University of Cagliari, Cagliari, Italy

    Enrica Santucci & Giuseppe Sergioli

Authors
  1. Enrica Santucci
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  2. Giuseppe Sergioli
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Editors and Affiliations

  1. International Center for Mathematical, Modeling in Physics and Cognitive Science, Linnaeus University, Växjö, Sweden

    Andrei Khrennikov

  2. Department of Mathematics, Howard University, Washington, DC, USA

    Bourama Toni

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Santucci, E., Sergioli, G. (2018). Classification Problem in a Quantum Framework. In: Khrennikov, A., Toni, B. (eds) Quantum Foundations, Probability and Information. STEAM-H: Science, Technology, Engineering, Agriculture, Mathematics & Health. Springer, Cham. https://doi.org/10.1007/978-3-319-74971-6_16

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