Abstract
We present a generalization of the problem of pattern recognition to arbitrary probabilistic models. This version deals with the problem of recognizing an individual pattern among a family of different species or classes of objects which obey probabilistic laws which do not comply with Kolmogorov’s axioms. We show that such a scenario accommodates many important examples, and in particular, we provide a rigorous definition of the classical and the quantum pattern recognition problems, respectively. Our framework allows for the introduction of non-trivial correlations (as entanglement or discord) between the different species involved, opening the door to a new way of harnessing these physical resources for solving pattern recognition problems. Finally, we present some examples and discuss the computational complexity of the quantum pattern recognition problem, showing that the most important quantum computation algorithms can be described as non-Kolmogorovian pattern recognition problems.
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Notes
An orthomodular lattice \({\mathcal{L}}\), is an orthocomplemented lattice satisfying that for any a, b and c, if \(a\le c\), then \(a\vee (a^{\bot }\wedge c)=c\). We refer the reader to Kalmbach (1983) for a detailed exposition.
In the Hilbert space case, projection operators are in one to one correspondence to closed subspaces (thus, these notions are interchangeable). Representing “\(\vee\)” by the closure of the sum of two subspaces, “\(\wedge\)” by its intersection, “\((\ldots )^{\bot }\)” by taking the orthogonal complement of a given subspace and “\(\le\)” by subspace inclusion, it is possible to show that subspaces (and thus, projections) possess an orthomodular lattice structure.
Notice that these operators could be quantum effects without loss of generality.
In practical implementations, these states and the discrimination problem, could be restricted to a concrete space-time region.
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Acknowledgements
This work was partially supported by CONICET and UNLP (Argentina), 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, Bando 2012 and the FIRB project Structures and Dynamics of Knowledge and Cognition, Cagliari: F21J12000140001, founded by Italian Ministery of Education. The authors thank anonymous reviewers for useful comments.
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Holik, F., Sergioli, G., Freytes, H. et al. Pattern Recognition in Non-Kolmogorovian Structures. Found Sci 23, 119–132 (2018). https://doi.org/10.1007/s10699-017-9520-4
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DOI: https://doi.org/10.1007/s10699-017-9520-4