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Pattern recognition is considered as a mapping from the set of all inputs to the numbers 0 to 1. The inputs are treated as vectors. A topological group algebra over the vector space is described. The input is treated as avariable in a polynomial of that group algebra. A correspondence between inputs and numbers is established. This correspondence is used to prove that the polynomials in the algebra can represent a solution to any pattern recognition problem. When the coefficients of the polynomial are suitably chosen vectors, the natural topology of the input vector space is preserved. The importance of this approach as a basis for a completely general efficient parallel process, and practically realizable pattern recognizing machine is presented. The concept may be realized by a modular parallel process type of machinery.