Towards Higher-Degree Fuzzy Projection

Fuzzy projection is a mathematical operator inspired by the inverse fuzzy transform that is used to approximate functions. The fuzzy projection is designed such that the coefficients of the linear combination of the basis functions (fuzzy sets in a fuzzy partition) are optimized to obtain the best approximation of the functions from a global perspective, as opposed to the fuzzy transform, where the approximation focuses on fitting functions locally. The aim of this paper is to extend the fuzzy projection to a higher degree, similarly to the fuzzy transform, where the coefficients of the linear combination of the basis functions are expressed by polynomials. In this way, we can significantly improve the quality of the approximation by combining the settings of the fuzzy partition and the degree of polynomnials. In this paper, we show that a higher-order fuzzy projection can be computed using matrix calculus, leading to an easy algorithmization of the method. We also give its approximation properties and its applicability to discrete functions. The usefulness of higher-order fuzzy projection is demonstrated on two tasks, namely continuous function approximation and audio signal compression and decompression, where the results are compared with other relevant methods.