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A quantum-inspired version of the nearest mean classifier

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Abstract

We introduce a framework suitable for describing standard classification problems using the mathematical language of quantum states. In particular, we provide a one-to-one correspondence between real objects and pure density operators. This correspondence enables us: (1) to represent the nearest mean classifier (NMC) in terms of quantum objects, (2) to introduce a quantum-inspired version of the NMC called quantum classifier (QC). By comparing the QC with the NMC on different datasets, we show how the first classifier is able to provide additional information that can be beneficial on a classical computer with respect to the second classifier.

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Notes

  1. Hence, as a pattern is an object characterized by the knowledge of its features, analogously, in quantum mechanics a state of a physical system is represented by a density operator, characterized by the knowledge of its observables.

  2. In the standard pattern recognition theory, the symbol y is generally used to identify the label of the pattern. In this paper, for the sake of simplicity, we agree with a different notation.

  3. Let us remark that, in general, \(a^*\) and \(b^*\) do not represent true centroids, but centroids estimated on the training set.

  4. For the sake of the simplicity, from now on, we indicate \(\sum _{j=1}^C \mathrm{TP}_j\) with TP. Similarly for TN, FP, and FN.

  5. This make sense because it can be seen that for all the datasets we deal with, the classification error is very similar both with and without splitting training and test sets.

  6. Let us remark that there are some patterns correctly classified by the NMC which are neglected by the QC. On this basis, exploiting their complementarity, in principle it also makes sense to consider a combination of both classifiers.

  7. In this case, by combining QC and NMC together, the mean error decreases up to about \(0.247\ (\pm 4.280)\).

  8. Similarly to the Gaussian case, also for the Banana dataset, the NMC is able to correctly classify some points unclassified by the QC. Indeed, by considering the combination of both classifiers, the mean error can decrease up to 10%.

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Acknowledgements

This work has been partly supported by the project “Computational quantum structures at the service of pattern recognition: modeling uncertainty” (CRP-59872) funded by Regione Autonoma della Sardegna, L.R. 7/2007 (2012) and the FIRB project “Structures and Dynamics of Knowledge and Cognition” (F21J12000140001).

Funding This study was funded by Regione Autonoma della Sardegna, L.R. 7/2007, CRP-59872 (2012).

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Authors and Affiliations

  1. Università di Cagliari, Via Is Mirrionis 1, 09123, Cagliari, Italy

    Giuseppe Sergioli, Enrica Santucci, Luca Didaci & Roberto Giuntini

  2. Institute of Theoretical and Applied Informatics, Polish Academy of Sciences, Bałtycka 5, 44-100, Gliwice, Poland

    Jarosław A. Miszczak

Authors
  1. Giuseppe Sergioli
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  2. Enrica Santucci
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  3. Luca Didaci
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  4. Jarosław A. Miszczak
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  5. Roberto Giuntini
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Correspondence to Giuseppe Sergioli.

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This article does not contain any studies with human participants or animals performed by any of the authors.

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Communicated by A. Di Nola.

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Sergioli, G., Santucci, E., Didaci, L. et al. A quantum-inspired version of the nearest mean classifier. Soft Comput 22, 691–705 (2018). https://doi.org/10.1007/s00500-016-2478-2

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