An algorithm for the non-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.

Refined methods for the identifiability of tensors

We prove that the general tensor of size 2^n and rank k has a unique decomposition as the sum of decomposable tensors if k<= 0.9997 (2^n)/(n+1) (the constant 1 being the optimal value). Similarly, the general tensor of size 3^n and rank k has a unique decomposition as the sum of decomposable tensors if k<= 0.998 (3^n)/(2n+1) (the constant 1 being the optimal value). Some results of this flavor are obtained for tensors of any size, but the explicit bounds obtained are weaker.

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

We propose a new sufficient condition for verifying whether generic rank-r complex tensors of arbitrary order admit a unique decomposition as a linear combination of rank-1 tensors. A practical algorithm is proposed for verifying this condition, with which it was established that in all spaces of dimension less than 15000, with a few known exceptions, listed in the paper, generic identifiability holds for ranks up to one less than the generic rank of the space. This is the largest possible rank value for which generic identifiability can hold, except for spaces with a perfect shape. The algori…

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…

Grassmann secants, identifiability, and linear systems of tensors

For any irreducible non-degenerate variety $X\subset \mathbb{P}^r$, we give a criterion for the $(k,s)$-identifiability of $X$. If $k\leq s-1 <r$, then the $(k,s)$-identifiability holds for $X$ if and only if the $s$-identifiability holds for the Segre product $Seg(\mathbb{P}^k\times X)$. Moreover, if the $s$-th secant variety of $X$ is not defective and it does not fill the ambient space, then we can produce a family of pairs $(k,s)$ for which the $(k,s)$-identifiability holds for $X$.

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。

One example of general unidentifiable tensors

The identifiability of parameters in a probabilistic model is a crucial notion in statistical inference. We prove that a general tensor of rank 8 in C^3\otimes C^6\otimes C^6 has at least 6 decompositions as sum of simple tensors, so it is not 8-identifiable. This is the highest known example of balanced tensors of dimension 3, which are not k-identifiable, when k is smaller than the generic rank.

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。

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

We study phase retrieval from magnitude measurements of an unknown signal as an algebraic estimation problem. Indeed, phase retrieval from rank-one and more general linear measurements can be treated in an algebraic way. It is verified that a certain number of generic rank-one or generic linear measurements are sufficient to enable signal reconstruction for generic signals, and slightly more generic measurements yield reconstructability for all signals. Our results solve a few open problems stated in the recent literature. Furthermore, we show how the algebraic estimation problem can be solved…

Subtracting a best rank-1 approximation may increase tensor rank

It has been shown that a best rank-R approximation of an order-k tensor may not exist when R>1 and k>2. This poses a serious problem to data analysts using tensor decompositions. It has been observed numerically that, generally, this issue cannot be solved by consecutively computing and subtracting best rank-1 approximations. The reason for this is that subtracting a best rank-1 approximation generally does not decrease tensor rank. In this paper, we provide a mathematical treatment of this property for real-valued 2x2x2 tensors, with symmetric tensors as a special case. Regardless of the symm…

Gaps in the pairs (border rank,symmetric rank) for symmetric tensors

Fix integers $m \ge 2$, $s\ge 5$ and $d\ge 2s+2$. Here we describe the possible symmetric tensor ranks $\le 2d+s-7$ of all symmetric tensors (or homogeneous degree $d$ polynomials) in $m+1$ variables with border rank $s$.

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.

A comparison of different notions of ranks of symmetric tensors

We introduce various notions of rank for a symmetric tensor, namely: rank, border rank, catalecticant rank, generalized rank, scheme length, border scheme length, extension rank and smoothable rank. We analyze the stratification induced by these ranks. The mutual relations between these stratifications, allow us to describe the hierarchy among all the ranks. We show that strict inequalities are possible between rank, border rank, extension rank and catalecticant rank. Moreover we show that scheme length, generalized rank and extension rank coincide.

Ranks of tensors and a generalization of secant varieties

We introduce subspace rank as a tool for studying ranks of tensors and X-rank more generally. We derive a new upper bound for the rank of a tensor and determine the ranks of partially symmetric tensors in C^2 \otimes C^b \otimes C^b. We review the literature from a geometric perspective.

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…

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

We study tensors in $\C^{m\times n\times l}$ whose border rank is $l$. We characterize the tensors in $\C^{3\times 3\times 4}$ and in $\C^{4\times 4\times 4}$ of border rank 4 at most.

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。

Computing symmetric rank for symmetric tensors

We consider the problem of determining the symmetric tensor rank for symmetric tensors with an algebraic geometry approach. We give algorithms for computing the symmetric rank for $2\times ... \times 2$ tensors and for tensors of small border rank. From a geometric point of view, we describe the symmetric rank strata for some secant varieties of Veronese varieties.

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.

Explicit tensors of border rank at least 2n-1

For odd n, I write down tensors in C^n\otimes C^n\otimes C^n of border rank 2n-1, showing the non-triviality of the Young-flattening equations of Landsberg-Ottaviani. I also study the border rank of the tensors of Alexeev et. al., showing the tensors their tensors T_{2^k}, despite having rank equal to 2^{k+1}-1, have border rank equal to 2^k, the minimum of any concise tensor. I also study the equations of Griesser.

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…

从签名张量学习路径

Learning Paths from Signature Tensors

本文通过张量分解、代数几何和数值优化方法，研究了张量的矩阵共轭问题，并针对随机分析中的逆问题，提出从第三阶签名张量恢复路径的方法，建立了路径的可识别性结果。

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。

Tensors of Nonnegative Rank Two

A nonnegative tensor has nonnegative rank at most 2 if and only if it is supermodular and has flattening rank at most 2. We prove this result, then explore the semialgebraic geometry of the general Markov model on phylogenetic trees with binary states, and comment on possible extensions to tensors of higher rank.

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.