Fast and stable recovery of Approximately low multilinear rank tensors from multi-way compressive measurements

Abstract

We introduce a reconstruction formula that allows one to recover an N -order tensor X ∈ R I1×···×IN from a reduced set of multi-way compressive measurements by exploiting its low multilinear rank structure. It is proved that, in the matrix case (N = 2), the proposed reconstruction is stable in the sense that the approximation error is proportional to the one provided by the best low-rank approximation, i.e X −X 2 ≤ K X − X 0 2 , where K is a constant and X 0 is the corresponding truncated SVD of X. We also present simulation results indicating that the same stable behavior is observed with higher order tensors (N > 2). In addition, it is shown that, an interesting property of multi-way measurements allows us to build the reconstruction based on compressive linear measurements of fibers taken only in two selected modes, independently of the tensor order N . Simulation results using real-world 2D and 3D signals are presented illustrating our results and comparing the reconstructions against the best low multilinear rank approximations and the reconstructions obtained by using the Kronecker-CS approach.

Stable, Robust, and Super Fast Reconstruction of Tensors Using Multi-Way Projections

“…In this work, we extend the ideas and results of our recent conference paper [24], providing a direct (i.e., analytical) reconstruction formula that allows us to recover a tensor from a set of multilinear projections that are obtained by multiplying the data tensor by a different sensing matrix in each mode. This model comes into scene Multilinear Non-iterative (direct) reconstruction naturally in many potential applications, for example, in the case of sensing 2D or 3D images by means of a separable operator as developed in [25], [26], [11], [12], i.e., by taking compressive measurements of columns, rows, etc.…”

Research on Three-Dimensional Imaging Method Using Tensor for Electrical Impedance Tomography (Eit)

“…In practical applications, the data tensor X is not in full rank, but has a good low linear-rank approximation. Therefore, an image signal can be written as [22]:…”

“…The quality of the reconstruction is quantified using peak signal to noise ratio (PNSR). The PNSR can be defined as PSNR (dB) = 20 log 10 max(X)/ X − X F (22) where X is a tensor that rearranges the actual pixel values, andX is the data reconstructed by the tensor reconstruction algorithm.…”