Triple Decomposition and Tensor Recovery of Third Order Tensors

采用带优化切片/特征选择的离散经验插值法的鲁棒低管秩张量恢复

Robust Low-Tubal-rank tensor recovery Using Discrete Empirical Interpolation Method with Optimized Slice/Feature Selection

该研究基于t积将离散经验插值法推广到三阶张量得到TDEIM，用于张量重要切片/特征选择，经理论和数值验证其近似精度优于现有方法，并可应用于监督分类任务。

Spectral Norm and Nuclear Norm of a Third Order Tensor

The spectral norm and the nuclear norm of a third order tensor play an important role in the tensor completion and recovery problem. We show that the spectral norm of a third order tensor is equal to the square root of the spectral norm of three positive semi-definite biquadratic tensors, and the square roots of the nuclear norms of those three positive semi-definite biquadratic tensors are lower bounds of the nuclear norm of that third order tensor. This provides a way to estimate and to evaluate the spectral norm and the nuclear norm of that third order tensor. Some upper and lower bounds fo…

An Orthogonal Equivalence Theorem for Third Order Tensors

In 2011, Kilmer and Martin proposed tensor singular value decomposition (T-SVD) for third order tensors. Since then, T-SVD has applications in low rank tensor approximation, tensor recovery, multi-view clustering, multi-view feature extraction, tensor sketching, etc. By going through the Discrete Fourier Transform (DFT), matrix SVD and inverse DFT, a third order tensor is mapped to an f-diagonal third order tensor. We call this a Kilmer-Martin mapping. We show that the Kilmer-Martin mapping of a third order tensor is invariant if that third order tensor is taking T-product with some orthogonal…

A tensor bidiagonalization method for higher-order singular value decomposition with applications

The need to know a few singular triplets associated with the largest singular values of third-order tensors arises in data compression and extraction. This paper describes a new method for their computation using the t-product. Methods for determining a couple of singular triplets associated with the smallest singular values also are presented. The proposed methods generalize available restarted Lanczos bidiagonalization methods for computing a few of the largest or smallest singular triplets of a matrix. The methods of this paper use Ritz and harmonic Ritz lateral slices to determine accurate…

谐均迭代加权最小二乘法用于低秩矩阵恢复

Harmonic Mean Iteratively Reweighted Least Squares for Low-Rank Matrix Recovery

本文提出一种新的迭代加权最小二乘算法，用于从不完整线性观测中恢复低秩矩阵，通过解决一系列低复杂度线性问题，实现低秩性促进，并在矩阵补全中表现出优异的全局收敛性、恢复概率和局部超线性收敛速度。

基于RIP的低管秩张量恢复性能保证

RIP-based performance guarantee for low-tubal-rank tensor recovery

提出正则化张量核范数最小化模型，通过定义张量受限等距性质，证明在t-RIP条件下可从少量线性噪声测量中稳定恢复三阶张量。