# An algorithm for the identifiability of rank-3 tensors

Pierpaola Santarsiero: An algorithm for the non-identifiability of rank-3 tensors

The main algorithm implements different functions in Macaulay2, aimed to understand if a concise tensor is a non-identifiable rank-3 tensor. To run the main algorithm one has to download also the file eqnTangential.m2 that implements equations for the tangential variety of a Segre variety given by the products of just projective lines (for which we refer to [Oe11] ).

Notation:

• We denote by $$X_{n_1,\dots,n_k}=\nu(Y_{n_1,\dots,n_k})$$ the Segre variety of the multiprojective space $$Y_{n_1,\dots,n_k}=\mathbb{P}^{n_1}\times \cdots \times \mathbb{P}^{n_k}$$.

• The tensor space $$\mathbb{C}^{n_1}\otimes \cdots \otimes \mathbb{C}^{n_k}$$ is denoted by $$\mathcal{T}_{n_1,\dots,n_k}$$ .

The algorithm is based on the classification Theorem 7.1 of [BBS], we briefly recall it here in the revised version of the Theorem (cf. Theorem 4.1).

A concise rank-3 tensor $$T\in \mathbb{C}^{n_1}\otimes \cdots \otimes \mathbb{C}^{n_k}$$ is identifiable except if $$T$$ is in one of the following families.

1. Matrix case The first trivial example of non-identifiable rank-3 tensors are $$3 \times 3$$ matrices, which is a very classical case.

2. Tangential case The tangential variety of a variety is the tangent developable of the variety itself. A point $$q$$ essentially lying on the tangential variety of the Segre $$X_{1,1,1}$$ is actually a point of the tangent space $$T_{[p]}X_{1,1,1}$$ for some $$p=u\otimes v \otimes w\in (\mathbb{C}^2)^{\otimes 3}$$. Therefore there exists some $$a,b,c\in \mathbb{C}^2$$ such that $$T$$ can be written as

$T=a\otimes v\otimes w+u\otimes b \otimes w+u\otimes v \otimes c$

and hence $$q$$ is actually non-identifiable.

3. Defective case We recall that the third secant variety of a Segre variety $$X_{n_1,\dots,n_k}$$ is defective if and only if $$(n_1,\dots ,n_k)=(1,1,1,1),(1,1,a)$$ with $$a\geq 3$$ (cf. Theorem 4.5 of [AOP]). The latter case do not play a role in the discussion and hence we can focus on the case $$k=4$$. By defectivity, the dimension of $$\sigma_3(X_{1,1,1,1})$$ is strictly smaller than the expected dimension and this proves that the generic element of $$\sigma_3(X_{1,1,1,1})$$ has an infinite number of rank-3 decompositions and therefore all the rank-3 tensor of this variety have an infinite number of decompositions.

4.5. Conic cases In this case one works with the Segre variety $$X_{2,1,1}$$ given by the image of a projective plane and two projective lines. Let $$Y_{2,1,1}=\mathbb{P}^2\times \mathbb{P}^1\times \mathbb{P}^1$$. Consider the Segre variety $$X_{1,1} \subset \mathbb{P}^3$$ given by the last two factors of $$Y_{2,1,1}$$ and take a hyperplane section which intersects $$X_{1,1}$$ in a conic $$\mathcal{C }$$. Let $$L_{\mathcal{C}}$$ be the Segre given by the product of the first factor $$\mathbb{P}^2$$ of $$Y_{2,1,1}$$ and the conic $$\mathcal{C}$$, therefore $$L_{\mathcal{C}}\subset X_{2,1,1}$$. The family of non-identifiable rank-3 tensors are points lying in the span of $$L_{\mathcal{C}}$$. In this case, the non-identifiability comes from the fact that the points on $$\langle \mathcal{C} \rangle$$ are not identifiable and the distinction between the two cases reflects the fact that the conic $$\mathcal{C}$$ can be either irreducible or reducible. The distinction between the two cases can be expressed as follows working in coordinates:

1. The non-identifiable tensor $$T\in\mathbb{C}^3\otimes \mathbb{C}^2\otimes \mathbb{C}^2$$ and there exists a basis $$\{u_1,u_2,u_3\}\subset\mathbb{C}^3$$ and a basis $$\{ v_1,v_2 \} \subset\mathbb{C}^2$$ such that $$T$$ can be written as

$T= u_1 \otimes v_1^2+u_2\otimes v_2^2 + u_3 \otimes (\alpha v_1+\beta v_2)^2,$

for some $$\alpha,\beta\neq 0$$;

1. The non-identifiable tensor $$T\in\mathbb{C}^3\otimes \mathbb{C}^2\otimes \mathbb{C}^2$$ and there exists a basis $$\{u_1,u_2,u_3\}\subset\mathbb{C}^3$$ and a basis $$\{ v_1,v_2 \} \subset \mathbb{C}^2$$ such that $$T$$ can be written as

$T= u_1 \otimes v_1\otimes \tilde{p}+u_2\otimes v_2 \otimes \tilde{p}+ u_3\otimes \tilde{q} \otimes w,$

for some $$\tilde{q}\in \langle v_1,v_2 \rangle$$, where $$\tilde{p},w \in \mathbb{C}^2$$ must be linearly independent;

6. General case The last family of non-identifiable rank-3 tensors relates the Segre variety $$X_{n_1,n_2,1^{k-2}}$$ that is the image of the multiprojective space $$Y_{n_1,n_2,1^{k-2}}=\mathbb{P}^{n_1}\times \mathbb{P}^{n_2}\times (\mathbb{P}^1)^{(k-2)}$$, where either $$k\geq 4$$ and $$n_1,n_2\in \{1,2\}$$ or $$k=3$$ and $$(n_1,n_2,n_3)\neq (2,1,1)$$. The non-identifiable rank-3 tensors of this case are as follows. Let $$Y':=\mathbb{P}^1\times \mathbb{P}^1\times \{ u_3\} \times \cdots \times \{ u_k\}$$ be a proper subset of $$Y_{n_1,n_2,1^{k-2}}$$, take $$q'$$ in the span of the Segre image of $$Y'$$ with the constrain that $$q'$$ is not an elementary tensor. Therefore $$q'$$ is a non-identifiable tensor of rank-2 since it can be seen as a $$2\times 2$$ matrix of rank-2. Let $$p\in X_{n_1,n_2,1^{k-2}}$$ be a rank-1 tensor taken outside the Segre image of $$Y'$$. Now any point $$q \in \langle \{q',p \} \rangle \setminus \{q' , p\}$$ is a rank-3 tensor and it is not identifiable since $$q'$$ has an infinite number of decompositions and each of these decompositions can be taken by considering $$p$$ together with a decomposition of $$q'$$.

For a coordinate description of this case, we take $$T \in \mathbb{C}^{m_1}\otimes \mathbb{C}^{m_2}\otimes (\mathbb{C}^2)^{k-2}$$, where $$k\geq 3$$, $$m_1,m_2\in \{2,3 \}$$ such that $$m_1+m_2+ (k-2)\geq 4$$. Moreover there exist distinct $$a_1,a_2\in \mathbb{C}^{m_1}$$, distinct $$b_1,b_2\in \mathbb{C}^{m_2}$$ and for all $$i\geq 3$$ there exists a basis $$\{u_i,\tilde{u}_i\}$$ of the $$i$$-th factor such that $$T$$ can be written as

$T= (a_1\otimes b_1+a_2\otimes b_2)\otimes u_3 \otimes \cdots \otimes u_k + a_3 \otimes b_3\otimes \tilde{u}_3 \otimes \cdots \otimes \tilde{u}_k,$

where if $$m_1=2$$ then $$a_3\in \langle a_1,a_2\rangle$$ otherwise $$a_1,a_2,a_3$$ are linearly independent. Similarly, if $$m_2=2$$ then $$b_3 \in \langle b_1,b_2\rangle$$, otherwise $$b_1,b_2,b_3$$ form a basis of the second factor.

References:

[AOP]
1. Abo, G. Ottaviani, C. Peterson, “Induction for secant varieties of Segre varieties”, Trans. Am. Math. Soc., 361:767-792, 2009.

[BBS]
1. Ballico, A. Bernardi, P. Santarsiero, “Identifiability of rank-3 tensors”, Mediterr. J. Math., 18:1-26, 2020.

[Oe11]
1. Oeding, “Set-theoretic defining equations of the tangential variety of the Segre variety.” J. Pure Appl. Algebra, 215.6:1516-1527 2011.

Project page created: 25/11/2021

Project contributors: Pierpaola Santarsiero

Software used: Macaulay2 (v1.19.1)

System setup used: MacBook Pro with macOS Ventura 13.0.1, Processor 2,6 GHz 6-Core Intel Core i7, Memory 16 GB 2667 MHz DDR4, Graphics Intel UHD Graphics 630 1536 MB.