  We have previously observed an irregular spatial distribution of photon
transmission through a photochromic crystal photoisomerized by a local optical
near-field excitation, manifesting complex branching processes via the
interplay of deformation of the material and near-field photon transfer
therein. Furthermore, by combining such naturally constructed complex photon
transmission with a simple photon detection protocol, Schubert polynomials, the
foundation of versatile permutation operations in mathematics, have been
generated. In this study, we demonstrate an order recognition algorithm
inspired by Schubert calculus using optical near-field statistics via
nanometre-scale photochromism. More specifically, by utilizing Schubert
polynomials generated via optical near-field patterns, we show that the order
of slot machines with initially unknown reward probability is successfully
recognized. We emphasize that, unlike conventional algorithms in the
literature, the proposed principle does not estimate the reward probabilities.
Instead, it exploits the inversion relations contained in the Schubert
polynomials. To quantitatively evaluate the impact of the Schubert polynomials
generated from an optical near-field pattern, order recognition performances
are compared with uniformly distributed and spatially strongly skewed
probability distributions, where the optical near-field pattern outperforms the
others. We found that the number of singularities contained in Schubert
polynomials and that of the given problem or considered environment exhibits a
clear correspondence, indicating that superior order recognition performances
may be attained if the singularity of the given problem is presupposed. This
study paves a new way toward nanophotonic intelligent devices and systems by
the interplay of complex natural processes and mathematical insights gained by
Schubert calculus.
