An algorithm for the 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:
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 \(\alpha,\beta\neq 0\);
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:
Abo, G. Ottaviani, C. Peterson, “Induction for secant varieties of Segre varieties”, Trans. Am. Math. Soc., 361:767-792, 2009.
Ballico, A. Bernardi, P. Santarsiero, “Identifiability of rank-3 tensors”, Mediterr. J. Math., 18:1-26, 2020.
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.
Corresponding author of this page: Pierpaola Santarsiero, pierpaola.santarsiero@mis.mpg.de