In the context of supervised tensor learning, preserving the structural information and exploiting the discriminative nonlinear relationships of tensor data are crucial for improving the performance of learning tasks. Based on tensor factorization theory and kernel methods, we propose a novel Kernelized Support Tensor Machine (KSTM) which integrates kernelized tensor factorization with maximum-margin criterion. Specifically, the kernelized factorization technique is introduced to approximate the tensor data in kernel space such that the complex nonlinear relationships within tensor data can be explored. Further, dual structural preserving kernels are devised to learn the nonlinear boundary between tensor data. As a result of joint optimization, the kernels obtained in KSTM exhibit better generalization power to discriminative analysis. The experimental results on real-world neuroimaging datasets show the superiority of KSTM over the state-of-the-art techniques.