Facial Recognition Using Tensor-Tensor Decompositions

A tensor is a multidimensional array. First-order tensors and second-order tensors can be viewed as vectors and matrices, respectively. Tensors of higher order, with the ability to include more information, appear more frequently nowadays in image and signal processing, data mining, biomedical engineering, and so on. With the recent work of Kilmer and Martin, familiar matrix-based factorizations in linear algebra can be extended in a straightforward way to third-order tensors based on their new tensor multiplication and concepts. Our method has an advantage over a popular tensor-based face recognition algorithm called TensorFaces, which is based on the higher-order SVD of an arrangement of the images in a database, in that it does not require a least squares solve for the coefficients. In this paper, we give a brief introduction to the new tensor framework and apply the induced tensor decompositions to the application of facial recognition. In the numerical results, we compare our new approaches with the traditional principal component analysis method based on matrix decomposition (also known as eigenfaces) and with TensorFaces.