Edoardo Ballico
We prove (without exceptions) the existence of irredundant tensor decompositions with the number of addenda equal to rank $+1$. We also discuss the existence of decompositions with more than the tensor rank terms, which are concise, while the original tensor is not concise.
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Decomposition loci of tensors
本文定义张量的分解轨迹为最小张量秩分解中的秩一张量集合，证明了位于任意Segre簇切向簇上的张量的分解轨迹仅排除切点，并显式计算了有限轨道张量空间中所有张量的分解轨迹。
On partial and generic uniqueness of block term tensor decompositions
给出了多线性秩（1，L_{1}，L_{1}），.的张量分解的通有唯一性的几个条件（1、L_{R}、L_{R}）术语。用几何语言证明了相关子空间簇的并不是切向弱亏损的。我们还通过证明相关子空间簇的连接不是亏损的，给出了块项张量分解的部分唯一性的条件。
Subtracting a best rank-1 approximation may increase tensor rank
已经证明，当R>1和k>2时，k阶张量的最佳秩R近似可能不存在。这给使用张量分解的数据分析师带来了严重的问题。已经在数值上观察到，通常，这个问题不能通过连续计算和减去最佳秩-1近似来解决。这样做的原因是减去最佳秩-1近似通常不会降低张量秩。在本文中，我们提供了一个数学处理这一性质的实值2x2 x2张量，对称张量作为一个特殊情况。无论对称性如何，我们证明了对于一般的2x2 x2张量（秩为2或3），减去最佳秩1近似会导致秩为3的张量，并且位于秩2和秩3集之间的边界上。因此，对于秩为2的典型张量，减去最佳秩1近似会增加张量秩。
Gaps in the pairs (border rank,symmetric rank) for symmetric tensors
固定整数$m \ge 2$、$s\ge 5$和$d\ge 2s+2$。在这里，我们描述了可能的对称张量秩$\le 2d+s-7$的所有对称张量（或齐次$d$多项式）在$m+1$变量与边界秩$s$。
We lower bound the rank of a tensor by a linear combination of the ranks of three of its unfoldings, using Sylvester's rank inequality. In a similar way, we lower bound the symmetric rank by a linear combination of the symmetric ranks of three unfoldings. Lower bounds on the rank and symmetric rank of tensors are important for finding counterexamples to Comon's conjecture. A real counterexample to Comon's conjecture is a tensor whose real rank and real symmetric rank differ. Previously, only one real counterexample was known. We divide the construction into three steps. The first step involves…
Hankel tensors are generalizations of Hankel matrices. This article studies the relations among various ranks of Hankel tensors. We give an algorithm that can compute the Vandermonde ranks and decompositions for all Hankel tensors. For a generic $n$-dimensional Hankel tensor of even order or order three, we prove that the the cp rank, symmetric rank, border rank, symmetric border rank, and Vandermonde rank all coincide with each other. In particular, this implies that the Comon's conjecture is true for generic Hankel tensors when the order is even or three. Some open questions are also posed.
We introduce the ``skew apolarity lemma'' and we use it to give algorithms for the skew-symmetric rank and the decompositions of tensors in {$\bigwedge^dV_{\mathbb{C}}$ with $d\leq 3$ and $\dim V_{\mathbb{C}} \leq 8$}. New algorithms to compute the rank and a minimal decomposition of a tritensor are also presented.
Shitov recently gave a counterexample to Comon's conjecture that the symmetric tensor rank and tensor rank of a symmetric tensor are the same. In this paper we show that an analog of Comon's conjecture for the G-stable rank introduced by Derksen is true: the symmetric G-stable rank and G-stable rank of a symmetric tensor are the same. We also show that the log-canonical threshold of a singularity is bounded by the G-stable rank of the defining ideal.
General Tensor Decomposition, Moment Matrices and Applications
本文所讨论的张量分解可以看作是矩阵的奇异值分解的推广。我们考虑一般的多线性和多齐次张量。我们展示了如何减少问题的截断矩矩阵问题，并给出了一个新的准Hankel矩阵的平坦扩张的标准。我们把这个标准连接到边界基地的交换特性。提出了一种新的算法。它适用于一般的多齐次张量，将J.J.西尔维斯特的方法扩展到二进制形式。一个例子说明了这种方法所涉及的代数运算以及如何从特征向量计算中恢复分解。
Toward Jordan Decompositions of Tensors
本文扩展Vinberg思想，将张量空间与作用李代数嵌入辅助代数，通过伴随算子建立张量不变量与Jordan分解的联系，并用于张量分解和量子信息中的轨道分类。
A comparison of different notions of ranks of symmetric tensors
本文引入了对称张量的秩的各种概念，即：秩、边秩、次秩、广义秩、概型长度、边概型长度、扩张秩和光滑秩。我们分析了这些职级引起的分层。这些等级之间的相互关系，使我们能够描述所有等级之间的等级关系。证明了秩、边秩、延拓秩和catalecticant秩之间的严格不等式是可能的。并且证明了格式长度、广义秩和扩张秩是一致的。
Rank-2 and rank-3 tensors are almost all identifiable with only few exceptions. We classify them all together with the dimensions and the structures of all the sets evincing the rank.
We present an algorithm aimed to recognize if a given tensor is a non-identifiable rank-3 tensor.
Border rank=rank for Kruskal tensors and a Kruskal's theorem for skew decompositions
该研究证明Kruskal张量的边界秩等于秩，给出交替张量的Kruskal条件以证明斜秩分解唯一性，还证明交替Kruskal张量的边界斜秩等于斜秩并给出最小斜秩分解算法。
This paper discusses the problem of symmetric tensor decomposition on a given variety $X$: decomposing a symmetric tensor into the sum of tensor powers of vectors contained in $X$. In this paper, we first study geometric and algebraic properties of such decomposable tensors, which are crucial to the practical computations of such decompositions. For a given tensor, we also develop a criterion for the existence of a symmetric decomposition on $X$. Secondly and most importantly, we propose a method for computing symmetric tensor decompositions on an arbitrary $X$. As a specific application, Vand…
In data processing and machine learning, an important challenge is to recover and exploit models that can represent accurately the data. We consider the problem of recovering Gaussian mixture models from datasets. We investigate symmetric tensor decomposition methods for tackling this problem, where the tensor is built from empirical moments of the data distribution. We consider identifiable tensors, which have a unique decomposition, showing that moment tensors built from spherical Gaussian mixtures have this property. We prove that symmetric tensors with interpolation degree strictly less th…
Gramian Tensor Decomposition via Semidefinite Programming
在本文中，我们研究了一个对称张量分解问题，Gramian分解，提出了一个秩最小化问题。我们研究放松的问题，并考虑的情况下，放松的解决方案是一个解决方案，原来的问题。在张量秩和阶的某些情况下，我们一般地证明了松弛的解在原解中是最优的。在其他情况下，我们提出了有趣的例子和方法，证明了这个问题的复杂性。
Unifying singular value decompositions of tensors via aligned orthogonality
本文研究基对齐双正交（bato）张量，证明其一般分解可识别、截断为临界低秩近似，确定其集合维数与闭包不可约分量的对应关系，并指出行列式等重要张量属于bato张量。
The geometry of rank decompositions of matrix multiplication I: 2x2 matrices
这是第一次在一系列文件的秩分解的矩阵乘法张量。在本文中，我们：建立了张量秩分解的一般事实，描述了寻找新的矩阵乘法分解的可能方法，给出了建立Strassen算法对称群的Burichenko定理的几何证明，并给出了Strassen分解族中两个特别好的子族。
Computing symmetric rank for symmetric tensors
利用代数几何方法研究了确定对称张量的对称张量秩的问题。我们给算法计算对称秩的$2\times ... \times 2$张量和小边界秩的张量。从几何的角度，我们描述了对称秩层的一些割线品种的Veronese品种。
These lecture notes are intended as an introduction to several notions of tensor rank and their connections to the asymptotic complexity of matrix multiplication. The latter is studied with the exponent of matrix multiplication, which will be expressed in terms of tensor (border) rank, (border) symmetric rank and the asymptotic rank of certain tensors. We introduce the multilinear rank of a tensor as well, deal with the concept of tensor equivalence and study prehomogeneous vector spaces with the castling transform. Moreover, we treat Apolarity Theory and use it to determine the symmetric rank…
It is shown that the relative distance in Frobenius norm of a real symmetric order-$d$ tensor of rank two to its best rank-one approximation is upper bounded by $\sqrt{1-(1-1/d)^{d-1}}$. This is achieved by determining the minimal possible ratio between spectral and Frobenius norm for symmetric tensors of border rank two, which equals $\left(1-{1}/{d}\right)^{(d-1)/{2}}$. These bounds are also verified for arbitrary real rank-two tensors by reducing to the symmetric case.
We discover a geometric property of the space of tensors of fixed multilinear (Tucker) rank. Namely, it is shown that real tensors of fixed multilinear rank form a minimal submanifold of the Euclidean space of tensors endowed with the Frobenius inner product. We also establish the absence of local extrema for linear functionals restricted to the submanifold of rank-one tensors, finding application in statistics.
We give algorithms to compute decompositions of a given polynomial, or more generally mixed tensor, as sum of rank one tensors, and to establish whether such a decomposition is unique. In particular, we present methods to compute the decomposition of a general plane quintic in seven powers, and of a general space cubic in five powers; the two decompositions of a general plane sextic of rank nine, and the five decompositions of a general plane septic. Furthermore, we give Magma implementations of all our algorithms.
Explicit tensors of border rank at least 2n-1
对于奇数n，我写下边界秩为2n-1的C^n \otimes C^n \otimes C^n中的张量，表明Landsberg-Ottaviani的Young平坦化方程的非平凡性。我还研究了Alexeev等张量的边界秩。例如，示出张量它们的张量T_{2^k}，尽管具有等于2^{k+1}-1的秩，但具有等于2 ^k的边界秩，这是任何简洁张量的最小值。我还研究了格里塞方程。
It has recently been shown that the tensor rank can be strictly submultiplicative under the tensor product, where the tensor product of two tensors is a tensor whose order is the sum of the orders of the two factors. The necessary upper bounds were obtained with help of border rank. It was left open whether border rank itself can be strictly submultiplicative. We answer this question in the affirmative. In order to do so, we construct lines in projective space along which the border rank drops multiple times and use this result in conjunction with a previous construction for a tensor rank drop…
Generating Polynomials and Symmetric Tensor Decompositions
本文研究对称张量分解，通过生成多项式描述其递归关系，并利用生成矩阵计算分解，探讨其性质及高效算法。
Constructions of Rank-Metric Codes of Small Tensor Rank
本文研究了秩度量码的张量秩与相关线性码参数之间的关系，提出了张量秩缺陷的概念，并利用代数几何码构造了具有小张量秩缺陷的秩度量码。
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