Symmetric Tensor Decompositions On Varieties

Tucker张量簇上的低秩优化

Low-rank optimization on Tucker tensor varieties

该研究探究Tucker张量簇的几何结构，提出可证明的梯度相关线搜索优化方法与秩自适应方法，实验表明其张量补全恢复性能优于现有最优方法，且能自动找到合适秩。

有界秩张量集合的去奇异化

Desingularization of bounded-rank tensor sets

本文针对有界秩张量集合的非光滑非凸问题，提出通过引入松弛变量将其去奇异化为光滑流形，从而将低秩优化转化为流形优化，并在张量补全实验中验证了方法的优越性。

A tensor SVD-like decomposition based on the semi-tensor product of tensors

In this paper, we define a semi-tensor product for third-order tensors. Based on this definition, we present a new type of tensor decomposition strategy and give the specific algorithm. This decomposition strategy actually generalizes the tensor SVD based on semi-tensor product. Due to the characteristic of semi-tensor product for compressing the data scale, we can therefore achieve data compression in this way. Numerical comparisons are given to show the advantages of this decomposition.

Triple Decomposition and Tensor Recovery of Third Order Tensors

In this paper, we introduce a new tensor decomposition for third order tensors, which decomposes a third order tensor to three third order low rank tensors in a balanced way. We call such a decomposition the triple decomposition, and the corresponding rank the triple rank. For a third order tensor, its CP decomposition can be regarded as a special case of its triple decomposition. The triple rank of a third order tensor is not greater than the middle value of the Tucker rank, and is strictly less than the middle value of the Tucker rank for an essential class of examples. These indicate that p…

The decompositions and positive semidefiniteness of fourth-order conjugate partial-symmetric tensors with applications

Conjugate partial-symmetric (CPS) tensor is a generalization of Hermitian matrices. For the CPS tensor decomposition some properties are presented. For real CPS tensors in particular, we note the subtle difference from the complex case of the decomposition. In addition to traditional decompositions in the form of the sum of rank-one tensors, we focus on the orthogonal matrix decomposition of CPS tensors, which inherits nice properties from decomposition of matrices. It then induces a procedure that reobtain the CPS decomposable property of CPS tensors. We also discuss the nonnegativity of the …

Low Rank Tensor Decompositions and Approximations

There exist linear relations among tensor entries of low rank tensors. These linear relations can be expressed by multi-linear polynomials, which are called generating polynomials. We use generating polynomials to compute tensor rank decompositions and low rank tensor approximations. We prove that this gives a quasi-optimal low rank tensor approximation if the given tensor is sufficiently close to a low rank one.