Membership Functional Analysis for Nonparametric Deep Models of Image Features
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 study fills this gap by providing an analytical nonparametric deep modeling solution based on the mathematical analysis of membership functions assigned to model variables. Our nonparametric approach is based on the concept of representing the unknown mappings (between input and output variables) through an infinite-dimensional Student-t type membership function. This concept is borrowed from Gaussian/Student-t processes based learning in kernel machines. The Student-t membership function based representation of a mapping is referred to as "Student-t fuzzy-mapping" in this study. The most significant feature of this study is to analytically derive the "optimal" mathematical expressions for membership functions using variational optimization. The study 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. The potential of the presented mathematical theory is demonstrated through numerous image classification experiments.
- M. Kumar and B. Freudenthaler, "Membership Functional Analysis for Nonparametric Deep Models of Image Features," IEEE Transactions on Fuzzy Systems, under-review.