  This paper introduces notions of the Drazin and the core-EP inverses on
tensors via M-product. We propose a few properties of the Drazin and the
core-EP inverse of tensors, as well as effective tensor-based algorithms for
calculating these inverses. In addition, definitions of composite generalized
inverses are presented in the framework of the M-product, including CMP, DMP,
and MPD inverse of tensors. Tensor-based higher-order Gauss-Seidel and
Gauss-Jacobi iterative methods are designed. Algorithms for these two iterative
methods to solve multilinear equations are developed. Certain multilinear
systems are solved using the Drazin inverse, core-EP inverses, and composite
generalized inverses such as CMP, DMP, and MPD inverse. A tensor
M-product-based regularization technique is applied to solve the color image
deblurring.
