Fuzzy Membership Functional Analysis for Nonparametric Deep Models of Image Features

Abstract

The application of fuzzy theory to deep learning is limited, 1) under the realm of deep neural networks, 2) to the parametric form of modeling, and 3) relying on gradient-descent-based numerical algorithms for optimization because of lack of analytical solutions. This article fills this gap by providing an analytical nonparametric deep modeling solution based on the mathematical analysis of membership functions assigned to model variables. The nonparametric approach is based on the concept of representing the unknown mappings (between input and output variables) through a fuzzy set with Student-t type membership function, such that the dimension of membership function increases with an increasing data size. This concept of function representation is referred to as Student-t fuzzy-mapping in this article. The most significant feature of this article is to analytically derive the mathematical expressions for membership functions (which quantify uncertainties regarding the values of variables) using variational optimization such that the degree-of-belongingness of given data to the considered data-model is maximized. This article focuses on the modeling of image features, where a layer of the deep-model first projects the feature vector onto a lower dimensional subspace, and then construct the output feature vector through Student- $t$ fuzzy-mappings. Numerous image classification experiments are provided to support the proposed approach.