P-Tensor Product in Compressed Sensing

Abstract

The dimension matching is a tough problem in the vector and matrix computations. In the traditional mode, there is only one way to calculate the angle between the 1-D plane and the 3-D vector, it is the projection. However, there are a number of lines on the plane, and taking only the projection to represent the plane is kind of a narrow choice. Furthermore, in the matrix multiplication, the dimension restriction is strict. In order to solve these problems, this paper defines a new model called ${P}$ -tensor product (PTP), which cannot only define the inner product of two vectors with unmatched dimensions but also give a new way to solve the problems in the matrix operations. Aiming at decreasing the large storage space of the random matrix in compressed sensing (CS), the PTP can reconstruct a high-dimensional matrix by using a matrix, which can be chosen as any kind of matrix. Similar with the traditional CS, we analyze some reconstruction conditions of PTP-CS such as, the spark, the coherence, and the restricted isometry property. The theorems proposed in this paper have a broad sense, and they possess a good universality for various tensor product CS methods. The experimental results demonstrate that our PTP-CS model can not only give more choices to the types of Kronecker matrix and decrease the storage space of the traditional CS but also maintain the considerable recovery performance. Besides, the proposed PTP-CS model can improve the signal transmission efficiency in the Internet of Things.