| In this thesis, we investigate a recently proposed tensor multiplication, then present new tensor concepts and decompositions based on the operation as well as their applications in facial recognition, image processing and optimization problems. Lower dimensional arrays, like vectors and matrices, have been the center of numerical study for centuries. Most real life applications use models based on vectors or matrices. However, with the ever-growing challenges of data storage and pursuit of higher accuracy, the study of tensor have gained a lot of attention in a variety of domains such as image and signal processing, data mining, biomedical engineering and so on. Tensors, as the higher dimensional generalization of matrices, are the natural fit for multilinear modeling. However, the study of tensors is still in the early stage. Most applications using tensor models still use traditional tensor multiplication and decompositions, which have some constraints and limitations. My study on tensors centers around a new tensor multiplication defined by M. Kilmer and C.D. Martin and can be mainly divided into three directions. In the first study, we investigate the use of the tensor SVD and tensor QR defined by M. Kilmer and C.D. Martin in the area of facial recognition. Specifically, we develop a PCA-like framework for this application. The application of the new tensor QR decomposition shows similar compression properties and recognition rates but has the advantage of fast updating/downdating. In the second study, we define a new nonnegative tensor decomposition (NTF) and develop algorithms for this decomposition. Inspired by the work of Homer F. Walker and Peng Ni, we adopt the Anderson acceleration in finding a NTF and get promising results. The third study is about defining new tensor nuclear norm with application in optimization problems such as those arising in color image deblurring. |
