Riordan Arrays as Structured Operators for Artificial Intelligence

원문 초록

Riordan arrays have traditionally been studied as infinite lower-triangular matrices that encode generating-function transformations and combinatorial relationships. This note proposes a complementary direction: using Riordan arrays as structured operators in artificial intelligence and data analysis. For a data sequence with generating function $A(z)$, a Riordan array $(g(z),f(z))$ produces the transformation \[ A(z)\longmapsto g(z)A(f(z)). \] This yields a parameterized family of causal linear operators that combine convolution, substitution, and multiscale mixing. The resulting structure may support parameter-efficient neural-network layers, variable-length sequence models, interpretable recurrent systems, invertible architectures, and models for hierarchical or recursively generated data. We explain what the infinite-dimensional viewpoint does and does not provide, outline several possible architectures, identify theoretical and numerical challenges, and propose an initial experimental program. The aim is not to claim that Riordan arrays are appropriate for all data, but to identify settings in which their algebraic structure supplies a useful inductive bias.