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

In the framework of multidimensional Compressed Sensing (CS), we introduce an analytical reconstruction formula that allows one to recover an $N$th-order $(I_1\times I_2\times \cdots \times I_N)$ data tensor $\underline{\mathbf{X}}$ from a reduced set of multi-way compressive measurements by exploiting its low multilinear-rank structure. Moreover, we show that, an interesting property of multi-way measurements allows us to build the reconstruction based on compressive linear measurements taken only in two selected modes, independently of the tensor order $N$. In addition, it is proved that, in the matrix case and in a particular case with $3$rd-order tensors where the same 2D sensor operator is applied to all mode-3 slices, the proposed reconstruction $\underline{\mathbf{X}}_\tau$ is stable in the sense that the approximation error is comparable to the one provided by the best low-multilinear-rank approximation, where $\tau$ is a threshold parameter that controls the approximation error. Through the analysis of the upper bound of the approximation error we show that, in the 2D case, an optimal value for the threshold parameter $\tau=\tau_0 > 0$ exists, which is confirmed by our simulation results. On the other hand, our experiments on 3D datasets show that very good reconstructions are obtained using $\tau=0$, which means that this parameter does not need to be tuned. Our extensive simulation results demonstrate the stability and robustness of the method when it is applied to real-world 2D and 3D signals. A comparison with state-of-the-arts sparsity based CS methods specialized for multidimensional signals is also included. A very attractive characteristic of the proposed method is that it provides a direct computation, i.e. it is non-iterative in contrast to all existing sparsity based CS algorithms, thus providing super fast computations, even for large datasets.Comment: Submitted to IEEE Transactions on Signal Processin

Low-rank approximate inverse for preconditioning tensor-structured linear systems

In this paper, we propose an algorithm for the construction of low-rank approximations of the inverse of an operator given in low-rank tensor format. The construction relies on an updated greedy algorithm for the minimization of a suitable distance to the inverse operator. It provides a sequence of approximations that are defined as the projections of the inverse operator in an increasing sequence of linear subspaces of operators. These subspaces are obtained by the tensorization of bases of operators that are constructed from successive rank-one corrections. In order to handle high-order tensors, approximate projections are computed in low-rank Hierarchical Tucker subsets of the successive subspaces of operators. Some desired properties such as symmetry or sparsity can be imposed on the approximate inverse operator during the correction step, where an optimal rank-one correction is searched as the tensor product of operators with the desired properties. Numerical examples illustrate the ability of this algorithm to provide efficient preconditioners for linear systems in tensor format that improve the convergence of iterative solvers and also the quality of the resulting low-rank approximations of the solution