Identifiability of rank-3 tensors

Tangential weak defectiveness and generic identifiability

We investigate the uniqueness of decomposition of general tensors $T\in {\mathbb C}^{n_1+1}\otimes\cdots\otimes{\mathbb C}^{n_r+1}$ as a sum of tensors of rank $1$. This is done extending the theory developed in a previous paper by the second author to the framework of non twd varieties. In this way we are able to prove the non generic identifiability of infinitely many partially symmetric tensors.

张量可识别性的精化方法

Refined methods for the identifiability of tensors

本文证明了当k<= 0.9997（2^n）/（n+1）（常数1为最优值）时，大小为2^n，秩为k的广义张量有唯一分解为可分张量之和.类似地，如果k<= 0.998（3^n）/（2n+1）（常数1是最优值），则大小为3^n且秩为k的一般张量具有唯一分解，即可分张量之和。对于任意大小的张量，得到了这种风格的一些结果，但得到的显式界较弱。

复张量通有和低秩特定可识别性的一种算法

An algorithm for generic and low-rank specific identifiability of complex tensors

本文提出了一个新的充分条件，用于证明任意阶的一般秩r复张量是否允许唯一分解为秩1张量的线性组合.提出了一个实用的算法来验证这个条件，它是建立在所有的空间尺寸小于小于15000，与少数已知的例外，在文件中列出的，一般的可识别性举行的行列小于的空间的一般秩。这是类属可识别性所能保持的最大可能秩值，除了具有完美形状的空间。该算法还可以验证给定特定rank-r分解的可识别性，前提是它可以被证明对应于r阶正割簇的非奇异点。对于足够小的秩，这仍然提高了特定的可识别性的已知界限，这种多样性的一些局部方程是已知的，使我们能够验证此属性。作为我们方法的一个特殊例子，我们证明了秩为7的特定5x 5x 5张量的可识别性，这不能用最近在[I]中提供的条件来处理。Domanov和L. De Lathauwer，关于三阶张量的正则多进分解的唯一性--第二部分：整体分解的唯一性，SIAM J.矩阵分析。34（3），2013]。最后，我们还提出了一个令人惊讶的新一类弱缺陷Segre品种，但原来承认一个通用的独特的分解。

Effective criteria for specific identifiability of tensors and forms

In applications where the tensor rank decomposition arises, one often relies on its identifiability properties for interpreting the individual rank-$1$ terms appearing in the decomposition. Several criteria for identifiability have been proposed in the literature, however few results exist on how frequently they are satisfied. We propose to call a criterion effective if it is satisfied on a dense, open subset of the smallest semi-algebraic set enclosing the set of rank-$r$ tensors. We analyze the effectiveness of Kruskal's criterion when it is combined with reshaping. It is proved that this cr…

Effective identifiability criteria for tensors and polynomials

A tensor $T$, in a given tensor space, is said to be $h$-identifiable if it admits a unique decomposition as a sum of $h$ rank one tensors. A criterion for $h$-identifiability is called effective if it is satisfied in a dense, open subset of the set of rank $h$ tensors. In this paper we give effective $h$-identifiability criteria for a large class of tensors. We then improve these criteria for some symmetric tensors. For instance, this allows us to give a complete set of effective identifiability criteria for ternary quintic polynomial. Finally, we implement our identifiability algorithms in M…

On generic identifiability of symmetric tensors of subgeneric rank

We prove that the general symmetric tensor in $S^d {\mathbb C}^{n+1}$ of rank r is identifiable, provided that r is smaller than the generic rank. That is, its Waring decomposition as a sum of r powers of linear forms is unique. Only three exceptional cases arise, all of which were known in the literature. Our original contribution regards the case of cubics ($d=3$), while for $d\ge 4$ we rely on known results on weak defectivity by Ballico, Ciliberto, Chiantini, and Mella.

Identifiability beyond Kruskal's bound for symmetric tensors of degree 4

We show how methods of algebraic geometry can produce criteria for the identifiability of specific tensors that reach beyond the range of applicability of the celebrated Kruskal criterion. More specifically, we deal with the symmetric identifiability of symmetric tensors in Sym$^4(\mathbb{C}^{n+1})$, i.e., quartic hypersurfaces in a projective space $\mathbb{P}^n$, that have a decomposition in 2n+1 summands of rank 1. This is the first case where the reshaped Kruskal criterion no longer applies. We present an effective algorithm, based on efficient linear algebra computations, that checks if t…

格拉斯曼割线、可识别性和张量的线性系

Grassmann secants, identifiability, and linear systems of tensors

对任意不可约非退化簇$X\subset \mathbb{P}^r$，给出了$X$的$(k,s)$ -可识别性的一个判别准则.如果$k\leq s-1 <r$，则当且仅当$s$ -可识别性对Segre产品$Seg(\mathbb{P}^k\times X)$成立时，$(k,s)$ -可识别性对$X$成立。此外，如果$X$的第$s$个正割簇不是亏损的，并且它不填充周围空间，那么我们可以产生一族对$(k,s)$，其中$(k,s)$可识别性对$X$成立。

Grassmannian割线簇的不可识别性与奇异轨迹

Identifiability and singular locus of secant varieties to Grassmannians

本研究利用SL(V)-簇结构确定了Grassmannian线割线簇的奇异轨迹，并解决了其中点的不可识别性与切向不可识别性问题，进而确定了第二Terracini轨迹。

张量分解与完美可辨识性的同伦技术

Homotopy techniques for tensor decomposition and perfect identifiability

本文利用同伦技术求解一般复张量的所有极小分解，并基于计算实验与代数几何证明格式(3,4,5)和(2,2,2,3)为新的通用可辨识性案例。

On the dimension of contact loci and the identifiability of tensors

Let $X\subset \mathbb{P}^r$ be an integral and non-degenerate variety. Set $n:= \dim (X)$. We prove that if the $(k+n-1)$-secant variety of $X$ has (the expected) dimension $(k+n-1)(n+1)-1<r$ and $X$ is not uniruled by lines, then $X$ is not $k$-weakly defective and hence the $k$-secant variety satisfies identifiability, i.e. a general element of it is in the linear span of a unique $S\subset X$ with $\sharp (S) =k$. We apply this result to many Segre-Veronese varieties and to the identifiability of Gaussian mixtures $G_{1,d}$. If $X$ is the Segre embedding of a multiprojective space we prove …

Waring decompositions and identifiability via Bertini and Macaulay2 software

Starting from our previous papers [AGMO] and [ABC], we prove the existence of a non-empty Euclidean open subset whose elements are polynomial vectors with 4 components, in 3 variables, degrees, respectively, 2,3,3,3 and rank 6, which are not identifiable over $ \mathbb{C} $ but are identifiable over $ \mathbb{R} $. This result has been obtained via computer-aided procedures suitably adapted to investigate the number of Waring decompositions for general polynomial vectors over the fields of complex and real numbers. Furthermore, by means of the Hessian criterion ([COV]), we prove identifiabilit…

On the identifiability of ternary forms

We describe a new method to determine the minimality and identifiability of a Waring decomposition $A$ of a specific form (symmetric tensor) $T$ in three variables. Our method, which is based on the Hilbert function of $A$, can distinguish between forms in the span of the Veronese image of $A$, which in general contains both identifiable and not identifiable points, depending on the choice of coefficients in the decomposition. This makes our method applicable for all values of the length $r$ of the decomposition, from $2$ up to the generic rank, a range which was not achievable before. Though …

On the description of identifiable quartics

In this paper we study the identifiability of specific forms (symmetric tensors), with the target of extending recent methods for the case of $3$ variables to more general cases. In particular, we focus on forms of degree $4$ in $5$ variables. By means of tools coming from classical algebraic geometry, such as Hilbert function, liaison procedure and Serre's construction, we give a complete geometric description and criteria of identifiability for ranks $\geq 9$, filling the gap between rank $\leq 8$, covered by Kruskal's criterion, and $15$, the rank of a general quartic in $5$ variables. For …

Hadamard-Hitchcock分解：可辨识性与计算

Hadamard-Hitchcock decompositions: identifiability and computation

本研究确立了多维数组的Hadamard-Hitchcock分解的通用可辨识性，提出借助现有张量秩分解算法的计算方法，实验验证了其性能与精度，可用于受限玻尔兹曼机相关的概率分布编码。

一般不可识别张量的一个例子

One example of general unidentifiable tensors

概率模型中参数的可辨识性是统计推断中的一个重要概念。证明了C^3 \otimes C^6 \otimes C^6中秩为8的一般张量至少有6个分解为单张量之和，因此它不是8-可辨识的.这是已知的最高的三维平衡张量的例子，当k小于一般秩时，它不是k-可识别的。

关于同时 Waring 可识别性猜想的一个反例

A counterexample to a conjecture on simultaneous Waring identifiability

本文通过计算机辅助程序构造反例，证明在 $d=4$ 时多项式向量的同时 Waring 分解不唯一，推翻了关于同时 Waring 可识别性完全分类的猜想。

平方和的广义可辨识性

Generalized identifiability of sums of squares

本文研究偶次齐次多项式表示为平方和时的分解轨道计数问题，给出泛型与特定 O(r)-可辨识性的充分条件，并证明三元形式具有泛型 O(r)-可辨识性。

相位恢复和可辨识门限显式反演的代数方法

The Algebraic Approach to Phase Retrieval and Explicit Inversion at the Identifiability Threshold

我们研究相位恢复从一个未知信号的幅度测量作为一个代数估计问题。实际上，从秩一和更一般的线性测量的相位恢复可以以代数方式处理。验证了一定数量的通用秩一或通用线性测量足以实现通用信号的信号重构，并且稍微更通用的测量产生所有信号的可重构性。我们的研究结果解决了一些开放的问题，在最近的文献中。此外，我们展示了如何代数估计问题可以解决的封闭形式的代数估计技术，称为理想回归，提供非渐近的成功保证。

减去最佳秩-1近似可以增加张量秩

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。

对称张量对（边界秩，对称秩）中的间隙

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$s$。

Lower bounds on the rank and symmetric rank of real tensors

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 tensor decompositions and ranks

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.

Tensor decompositions in rank +1

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.

G-stable rank of symmetric tensors and log canonical threshold

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.

对称张量秩的不同概念的比较

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。

张量的秩与割线簇的推广

Ranks of tensors and a generalization of secant varieties

我们引入子空间秩作为研究张量秩和X秩的工具。本文给出了张量秩的一个新的上界，并确定了C^2 \otimes C^b \otimes C^b中部分对称张量的秩.我们从几何的角度回顾文献。

Identifiability of linear compartmental models: the singular locus

This work addresses the problem of identifiability, that is, the question of whether parameters can be recovered from data, for linear compartmental models. Using standard differential algebra techniques, the question of whether a given model is generically locally identifiable is equivalent to asking whether the Jacobian matrix of a certain coefficient map, arising from input-output equations, is generically full rank. A natural next step is to study the set of parameter values where the Jacobian matrix drops in rank, which we refer to as the locus of non-identifiable parameter values, or, fo…

关于$\C^{m\times n\times l}$中边秩$l$的张量

On tensors of border rank $l$ in $\C^{m\times n\times l}$

我们研究边界秩为$l$的$\C^{m\times n\times l}$中的张量。我们刻画了边界秩最多为4的$\C^{3\times 3\times 4}$和$\C^{4\times 4\times 4}$中的张量。

过完备独立成分分析的可识别性

Identifiability of overcomplete independent component analysis

该研究将独立成分分析的可识别性理论推广至源数多于观测数的过完备场景，给出充要条件与通用混合下的判定依据，并基于此设计算法在合成及真实数据集上验证有效性。

Kruskal张量的边界秩等于秩，以及斜分解的Kruskal定理

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。

计算对称张量的对称秩

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。

Tensor Rank and Complexity

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…

Maximum relative distance between real rank-two and rank-one tensors

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.

Minimality of tensors of fixed multilinear rank

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.

边界秩至少为2n-1的显式张量

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。

Border rank is not multiplicative under the tensor product

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…

Best rank-$k$ approximations for tensors: generalizing Eckart-Young

Given a tensor $f$ in a Euclidean tensor space, we are interested in the critical points of the distance function from $f$ to the set of tensors of rank at most $k$, which we call the critical rank-at-most-$k$ tensors for $f$. When $f$ is a matrix, the critical rank-one matrices for $f$ correspond to the singular pairs of $f$. The critical rank-one tensors for $f$ lie in a linear subspace $H_f$, the critical space of $f$. Our main result is that, for any $k$, the critical rank-at-most-$k$ tensors for a sufficiently general $f$ also lie in the critical space $H_f$. This is the part of Eckart-Yo…

小张量秩的秩度量码构造

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。

非负秩二张量

Tensors of Nonnegative Rank Two

一个非负张量的非负秩至多为2当且仅当它是超模的，且平坦秩至多为2。我们证明了这一结果，然后探讨一般马尔可夫模型的系统发育树与二进制状态的半代数几何，并评论可能的扩展到更高的秩张量。

对称张量与对称张量秩

Symmetric tensors and symmetric tensor rank

本文研究对称张量的性质，特别是其分解为对称外积向量和的形式，并探讨秩与对称秩的关系以及一般对称秩的已知结果。

何时减去秩一近似会降低张量秩？

When does subtracting a rank-one approximation decrease tensor rank?

本文研究减去临界秩一近似后张量秩降低的条件，构造了相关张量簇，并揭示其与秩一簇瓶颈点及超行列式奇异轨迹的联系。

The G-stable rank for tensors

We introduce the $G$-stable rank of a higher order tensors over perfect fields. The $G$-stable rank is related to the Hilbert-Mumford criterion for stability in Geometric Invariant Theory. We will relate the $G$-stable rank to the tensor rank and slice rank. For numerical applications, we express the $G$-stable rank as a solution to an optimization problem. Over the field ${\mathbb F}_3$ we discuss an application to the Cap Set Problem.