On the uniqueness of the canonical polyadic decomposition of third-order tensors. I: Basic results and uniqueness of one factor matrix (Q2866214)

Let \(\mathbb{F}\) denote the real or complex field and let \(I,J,K\) and \(R\) be positive integers. A (third-order) tensor \(\mathcal{T}\) is an \(I\times J\times K\) array \([t_{ijk}]\) over \(\mathbb{F}\). In particular, given nonzero vectors \(a\in\mathbb{F}^{I}\), \(b\in\mathbb{F}^{J}\) and \(c\in\mathbb{F}^{K}\), the outer product \(a\circ b\circ c\) is a tensor whose \((i,j,k)\)th entry is the product of the \(i\)th entry of \(a\), the \(j\)th entry of \(b\) and the \(k\)th entry of \(c\). We say that \(a\circ b\circ c\) is a tensor of rank \(1\) (note that \(a'\circ b'\circ c'=a\circ b\circ c\) only if corresponding factors are scalar multiples of one another). Every third-order tensor can be written as \(\mathcal{T=}\) \(\sum_{r=1}^{R}a_{r}\circ b_{r}\circ c_{r}\) as a sum of tensors of rank \(1\), and the latter is called a polyadic decomposition (PD). We also write this PD in the form \(\mathcal{T=}\left[ A,B,C\right] _{R}\) there \(A,B\) and \(C\) are matrices whose columns are \(a_{1},\dots,a_{R}\); \(b_{1},\dots,b_{R}\) and \(c_{1},\dots,c_{R}\), respectively. The least value \(r_{T}\) of \(R\) for which \(\mathcal{T}\) has a PD is called the rank of \(\mathcal{T}\) and a PD with \(R=r_{T}\) is called a canonical polyadic decomposition (CPD) of \(\mathcal{T}\). Since the ordering of the summands in \(T\) is irrelevant, we say that a PD \([A,B,C]_{R}\) for \(\mathcal{T}\) is unique if, whenever \(\mathcal{T=}\left[ A',B',C'\right] _{R}\), there exists an \(R\times R\) permutation matrix \(\Pi\) and nonsingular \(R\times R\) diagonal matrices \(\Lambda_{A},\Lambda_{B}\) and \(\Lambda_{C}\) such that \(A'=A\Pi\Lambda_{A}\), \(B'=B\Pi\Lambda_{B}\) and \(C^{\prime }=C\Pi\Lambda_{C}\); more generally, \(C\) is unique when the last of these three conditions holds. CPDs have applications in signal processing, data analysis, etc. (see, for example [\textit{T. G. Kolda} and \textit{B. W. Bader}, SIAM Rev. 51, No. 3, 455--500 (2009; Zbl 1173.65029)]).NEWLINENEWLINEAn important question is: Is a CPD for \(\mathcal{T}\) unique? Or, more generally, is the third factor of the CPD unique? There is an extensive literature on this topic, and a fundamental result is the following. For any matrix \(X\) let \(k_{X}\) be the largest integer \(k\) such that every set of \(k\) columns of \(X\) is linearly independent. Kruskal's theorem states: If \(\mathcal{T}=[A,B,C]_{R}\) with \(k_{A}+k_{B}+k_{C}\geq2R+2\), then \(R=r_{\mathcal{T}}\) and the CPD of \(\mathcal{T}\) is unique (see [\textit{J. B. Kruskal}, Linear Algebra Appl. 18, 95--138 (1977; Zbl 0364.15021)]).NEWLINENEWLINEIn the first of the papers under review the authors survey results similar to Kruskal's and develop stronger versions of some of them. There are numerous results under many different hypotheses, but here is an example. The condition \((U_{m})\) holds for \(A\in\mathbb{F}^{I\times R}\) and \(B\in\mathbb{F}^{J\times R}\) if, for every diagonal \(R\times R\) matrix \(D\), \(\mathrm{rank}(ADB^{T})\leq m-1\) implies \(\mathrm{rank}(D)\leq m-1\). The authors prove: (\((U_{2})\) holds for \(A\) and \(B\)) \(\iff\) (\(r_{\mathcal{T}}=R\) and \(C\) is unique) \(\iff\) (\(\mathcal{T=} \left[ A,B,C\right] _{R}\) is a unique CPD). A similar result is: for every PD \(\mathcal{T}=[A,B,C]_{R}\) for which \(k_{C}\geq1\), \(m:=R-\mathrm{rank}(C)+2\) \(\leq\) \(\min(I,J)\) and \((U_{m})\) holds for \(A\) and \(B\), we have \(r_{\mathcal{T}}=R\) and \(C\) is unique.NEWLINENEWLINEThe second paper is a continuation of this work. In particular, the authors consider uniqueness theorems for ``generic'' tensors of given dimensions and rank; that is, theorems which hold for all but a set of measure \(0\) of triples \((A,B,C)\), see, for example ([\textit{V. Strassen}, Linear Algebra Appl. 52--53, 645--685 (1983; Zbl 0514.15018)] and [\textit{L. Chiantini} and \textit{G. Ottaviani}, SIAM J. Matrix Anal. Appl. 33, No. 3, 1018--1037 (2012; Zbl 1263.14053)]). For example, suppose that \(C\in\mathbb{F}^{K\times R}\) is nonzero, and that for some \(A_{0}\in\mathbb{F}^{I\times R}\) and \(B_{0}\in\mathbb{F}^{J\times R}\) the pair \((A_{0},B_{0})\) satisfies \((U_{m})\) with \(m:=R-\mathrm{rank}(C)+2.\) Then generically \(\mathcal{T}=[A,B,C]_{R}\) has rank \(R\) and the third factor is unique. Moreover, if \(C\) has rank \(k_{C}\), then generically the CPD of \(\mathcal{T}\) is unique. The paper concludes with numerical information about uniqueness of the CPD for generic tensors with small values of the indices.