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.

该研究提出了一种基于低秩结构的多维压缩感知重构公式，证明了在只采用两种选择的测量方式下可以恢复目标数据张量，讨论了阈值参数对误差控制的影响，并给出了一种快速计算的非迭代算法，实验证明该方法的性能稳定且鲁棒性强。

使用多维投影稳定、健壮且超快速重构张量

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

Iterative thresholding algorithms are well-suited for high-dimensional
problems in sparse recovery and compressive sensing. The performance of this
class of algorithms depends heavily on the tuning of certain threshold
parameters. In particular, both the final reconstruction error and the
convergence rate of the algorithm crucially rely on how the threshold parameter
is set at each step of the algorithm. In this paper, we propose a
parameter-free approximate message passing (AMP) algorithm that sets the
threshold parameter at each iteration in a fully automatic way without either
having an information about the signal to be reconstructed or needing any
tuning from the user. We show that the proposed method attains both the minimum
reconstruction error and the highest convergence rate. Our method is based on
applying the Stein unbiased risk estimate (SURE) along with a modified gradient
descent to find the optimal threshold in each iteration. Motivated by the
connections between AMP and LASSO, it could be employed to find the solution of
the LASSO for the optimal regularization parameter. To the best of our
knowledge, this is the first work concerning parameter tuning that obtains the
fastest convergence rate with theoretical guarantees.

本文提出一种无需用户调整阈值参数的自动方式，采用修改过的梯度下降算法和 Stein 无偏风险估计，实现高稀疏恢复和压缩感知算法中的最小重构误差和最高收敛速率，为参数调整领域的最快收敛率理论保障提供了解决方案。