New Results on Classification Modeling of Noisy Tensor Datasets: A Fuzzy Support Tensor Machine Dual Model

Abstract

In this article, classification problems for a class of tensor datasets with a noisy environment are investigated. To address such issues, a novel fuzzy support tensor machine (FSTM) dual model with robustness is established. First, for each input sample in the noisy tensor dataset, we define three kinds of fuzzy membership functions, such as linear, cosine, and exponential forms. In particular, the reconstruction process from one-dimensional (1-D) vector data to third-order tensor data is also derived in the Appendix. Second, the original optimization model of an FSTM on fuzzy membership is designed by constructing the vector pattern of the traditional support vector machine (SVM) models into a tensor pattern. Next, by introducing the Lagrangian multiplier method and tensor-Tucker decomposition method to the original FSTM model, an FSTM dual model without tensor inner product operation is obtained for the first time. Such a dual model with tensor-Tucker decomposition form can avoid conservativeness caused by the vectorization of tensor data in the traditional SVM model. Furthermore, an FSTM classifier is derived by the designed numerical algorithm, and the classification generalization error bound of the FSTM model with a general form is developed. It is worth noting that a linear least-squares FSTM (LLS-FSTM) equation with tensor-Tucker decomposition is also designed in the Appendix to further reduce the slightly time-consuming problem of the solving the dual optimization model FSTM. Finally, two numerical examples are presented to verify the feasibility and validity of the derived FSTM classifier.