Tensor rank decomposition

The rank of a tensor depends on the field over which the tensor is decomposed. It is known that some real tensors may admit a complex decomposition whose rank is strictly less than the rank of a real decomposition of the same tensor. As an example,[8] consider the following real tensor

${\displaystyle {\mathcal {A}}=\mathbf {x} _{1}\otimes \mathbf {x} _{2}\otimes \mathbf {x} _{3}+\mathbf {x} _{1}\otimes \mathbf {y} _{2}\otimes \mathbf {y} _{3}-\mathbf {y} _{1}\otimes \mathbf {x} _{2}\otimes \mathbf {y} _{3}+\mathbf {y} _{1}\otimes \mathbf {y} _{2}\otimes \mathbf {x} _{3},}$

where ${\displaystyle \mathbf {x} _{i},\mathbf {y} _{j}\in \mathbb {R} ^{2}}$. The rank of this tensor over the reals is known to be 3, while its complex rank is only 2 because it is the sum of a complex rank-1 tensor with its complex conjugate, namely

${\displaystyle {\mathcal {A}}={\frac {1}{2}}({\bar {\mathbf {z} }}_{1}\otimes \mathbf {z} _{2}\otimes {\bar {\mathbf {z} }}_{3}+\mathbf {z} _{1}\otimes {\bar {\mathbf {z} }}_{2}\otimes \mathbf {z} _{3}),}$

where ${\displaystyle \mathbf {z} _{k}=\mathbf {x} _{k}+i\mathbf {y} _{k}}$.

In contrast, the rank of real matrices will never decrease under a field extension to ${\displaystyle \mathbb {C} }$: real matrix rank and complex matrix rank coincide for real matrices.

The generic rank ${\displaystyle r(I_{1},\ldots ,I_{M})}$ is defined as the least rank ${\displaystyle r}$ such that the closure in the Zariski topology of the set of tensors of rank at most ${\displaystyle r}$ is the entire space ${\displaystyle F^{I_{1}}\otimes \cdots \otimes F^{I_{M}}}$. In the case of complex tensors, tensors of rank at most ${\displaystyle r(I_{1},\ldots ,I_{M})}$ form a dense set ${\displaystyle S}$: every tensor in the aforementioned space is either of rank less than the generic rank, or it is the limit in the Euclidean topology of a sequence of tensors from ${\displaystyle S}$. In the case of real tensors, the set of tensors of rank at most ${\displaystyle r(I_{1},\ldots ,I_{M})}$ only forms an open set of positive measure in the Euclidean topology. There may exist Euclidean-open sets of tensors of rank strictly higher than the generic rank. All ranks appearing on open sets in the Euclidean topology are called typical ranks. The smallest typical rank is called the generic rank; this definition applies to both complex and real tensors. The generic rank of tensor spaces was initially studied in 1983 by Volker Strassen.[9]

As an illustration of the above concepts, it is known that both 2 and 3 are typical ranks of ${\displaystyle \mathbb {R} ^{2}\otimes \mathbb {R} ^{2}\otimes \mathbb {R} ^{2}}$ while the generic rank of ${\displaystyle \mathbb {C} ^{2}\otimes \mathbb {C} ^{2}\otimes \mathbb {C} ^{2}}$ is 2. Practically, this means that a randomly sampled real tensor (from a continuous probability measure on the space of tensors) of size ${\displaystyle 2\times 2\times 2}$ will be a rank-1 tensor with probability zero, a rank-2 tensor with positive probability, and rank-3 with positive probability. On the other hand, a randomly sampled complex tensor of the same size will be a rank-1 tensor with probability zero, a rank-2 tensor with probability one, and a rank-3 tensor with probability zero. It is even known that the generic rank-3 real tensor in ${\displaystyle \mathbb {R} ^{2}\otimes \mathbb {R} ^{2}\otimes \mathbb {R} ^{2}}$ will be of complex rank equal to 2.

The generic rank of tensor spaces depends on the distinction between balanced and unbalanced tensor spaces. A tensor space ${\displaystyle F^{I_{1}}\otimes \cdots \otimes F^{I_{M}}}$, where ${\displaystyle I_{1}\geq I_{2}\geq \cdots \geq I_{M}}$, is called unbalanced whenever

${\displaystyle I_{1}>1+\prod _{m=2}^{M}I_{m}-\sum _{m=2}^{M}(I_{m}-1),}$

and it is called balanced otherwise.

When the first factor is very large with respect to the other factors in the tensor product, then the tensor space essentially behaves as a matrix space. The generic rank of tensors living in an unbalanced tensor spaces is known to equal

${\displaystyle r(I_{1},\ldots ,I_{M})=\min \left\{I_{1},\prod _{m=2}^{M}I_{m}\right\}}$

almost everywhere. More precisely, the rank of every tensor in an unbalanced tensor space ${\displaystyle F^{I_{1}\times \cdots \times I_{M}}\setminus Z}$, where ${\displaystyle Z}$ is some indeterminate closed set in the Zariski topology, equals the above value.[10]

The expected generic rank of tensors living in a balanced tensor space is equal to

${\displaystyle r_{E}(I_{1},\ldots ,I_{M})=\left\lceil {\frac {\Pi }{\Sigma +1}}\right\rceil }$

almost everywhere for complex tensors and on a Euclidean-open set for real tensors, where

${\displaystyle \Pi =\prod _{m=1}^{M}I_{m}\quad {\text{and}}\quad \Sigma =\sum _{m=1}^{M}(I_{m}-1).}$

More precisely, the rank of every tensor in ${\displaystyle \mathbb {C} ^{I_{1}\times \cdots \times I_{M}}\setminus Z}$, where ${\displaystyle Z}$ is some indeterminate closed set in the Zariski topology, is expected to equal the above value.[11] For real tensors, ${\displaystyle r_{E}(I_{1},\ldots ,I_{M})}$ is the least rank that is expected to occur on a set of positive Euclidean measure. The value ${\displaystyle r_{E}(I_{1},\ldots ,I_{M})}$ is often referred to as the expected generic rank of the tensor space ${\displaystyle F^{I_{1}\times \cdots \times I_{M}}}$ because it is only conjecturally correct. It is known that the true generic rank always satisfies

${\displaystyle r(I_{1},\ldots ,I_{M})\geq r_{E}(I_{1},\ldots ,I_{M}).}$

The Abo–Ottaviani–Peterson conjecture[11] states that equality is expected, i.e., ${\displaystyle r(I_{1},\ldots ,I_{M})=r_{E}(I_{1},\ldots ,I_{M})}$, with the following exceptional cases:

• ${\displaystyle F^{(2m+1)\times (2m+1)\times 3}{\text{ with }}m=1,2,\ldots }$
• ${\displaystyle F^{(m+1)\times (m+1)\times 2\times 2}{\text{ with }}m=2,3,\ldots }$

In each of these exceptional cases, the generic rank is known to be ${\displaystyle r(I_{1},\ldots ,I_{m},\ldots ,I_{M})=r_{E}(I_{1},\ldots ,I_{M})+1}$. Note that while the set of tensors of rank 3 in ${\displaystyle F^{2\times 2\times 2\times 2}}$ is defective (13 and not the expected 14), the generic rank in that space is still the expected one, 4. Similarly, the set of tensors of rank 5 in ${\displaystyle F^{4\times 4\times 3}}$ is defective (44 and not the expected 45), but the generic rank in that space is still the expected 6.

The AOP conjecture has been proved completely in a number of special cases. Lickteig showed already in 1985 that ${\displaystyle r(n,n,n)=r_{E}(n,n,n)}$, provided that ${\displaystyle n\neq 3}$.[12] In 2011, a major breakthrough was established by Catalisano, Geramita, and Gimigliano who proved that the expected dimension of the set of rank ${\displaystyle s}$ tensors of format ${\displaystyle 2\times 2\times \cdots \times 2}$ is the expected one except for rank 3 tensors in the 4 factor case, yet the expected rank in that case is still 4. As a consequence, ${\displaystyle r(2,2,\ldots ,2)=r_{E}(2,2,\ldots ,2)}$ for all binary tensors.[13]

The maximum rank that can be admitted by any of the tensors in a tensor space is unknown in general; even a conjecture about this maximum rank is missing. Presently, the best general upper bound states that the maximum rank ${\displaystyle r_{\mbox{max}}(I_{1},\ldots ,I_{M})}$ of ${\displaystyle F^{I_{1}}\otimes \cdots \otimes F^{I_{M}}}$, where ${\displaystyle I_{1}\geq I_{2}\geq \cdots \geq I_{M}}$, satisfies

${\displaystyle r_{\mbox{max}}(I_{1},\ldots ,I_{M})\leq \min \left\{\prod _{m=2}^{M}I_{m},2\cdot r(I_{1},\ldots ,I_{M})\right\},}$

where ${\displaystyle r(I_{1},\ldots ,I_{M})}$ is the (least) generic rank of ${\displaystyle F^{I_{1}}\otimes \cdots \otimes F^{I_{M}}}$.[14] It is well-known that the foregoing inequality may be strict. For instance, the generic rank of tensors in ${\displaystyle \mathbb {R} ^{2\times 2\times 2}}$ is two, so that the above bound yields ${\displaystyle r_{\mbox{max}}(2,2,2)\leq 4}$, while it is known that the maximum rank equals 3.[8]

A rank-${\displaystyle s}$ tensor ${\displaystyle {\mathcal {A}}}$ is called a border tensor if there exists a sequence of tensors of rank at most ${\displaystyle r<s}$ whose limit is ${\displaystyle {\mathcal {A}}}$. If ${\displaystyle r}$ is the least value for which such a convergent sequence exists, then it is called the border rank of ${\displaystyle {\mathcal {A}}}$. For order-2 tensors, i.e., matrices, rank and border rank always coincide, however, for tensors of order ${\displaystyle \geq 3}$ they may differ. Border tensors were first studied in the context of fast approximate matrix multiplication algorithms by Bini, Lotti, and Romani in 1980.[15]

A classic example of a border tensor is the rank-3 tensor

${\displaystyle {\mathcal {A}}=\mathbf {u} \otimes \mathbf {u} \otimes \mathbf {v} +\mathbf {u} \otimes \mathbf {v} \otimes \mathbf {u} +\mathbf {v} \otimes \mathbf {u} \otimes \mathbf {u} ,\quad {\text{with }}\|\mathbf {u} \|=\|\mathbf {v} \|=1{\text{ and }}\langle \mathbf {u} ,\mathbf {v} \rangle \neq 1.}$

It can be approximated arbitrarily well by the following sequence of rank-2 tensors

${\displaystyle {\begin{aligned}{\mathcal {A}}_{m}&=m(\mathbf {u} +{\frac {1}{m}}\mathbf {v} )\otimes (\mathbf {u} +{\frac {1}{m}}\mathbf {v} )\otimes (\mathbf {u} +{\frac {1}{m}}\mathbf {v} )-m\mathbf {u} \otimes \mathbf {u} \otimes \mathbf {u} \\&=\mathbf {u} \otimes \mathbf {u} \otimes \mathbf {v} +\mathbf {u} \otimes \mathbf {v} \otimes \mathbf {u} +\mathbf {v} \otimes \mathbf {u} \otimes \mathbf {u} +{\frac {1}{m}}(\mathbf {u} \otimes \mathbf {v} \otimes \mathbf {v} +\mathbf {v} \otimes \mathbf {u} \otimes \mathbf {v} +\mathbf {v} \otimes \mathbf {v} \otimes \mathbf {u} )+{\frac {1}{m^{2}}}\mathbf {v} \otimes \mathbf {v} \otimes \mathbf {v} \end{aligned}}}$

as ${\displaystyle m\to \infty }$. Therefore, its border rank is 2, which is strictly less than its rank. When the two vectors are orthogonal, this example is also known as a W state.

It follows from the definition of a pure tensor that ${\displaystyle {\mathcal {A}}=\mathbf {a} _{1}\otimes \mathbf {a} _{2}\otimes \cdots \otimes \mathbf {a} _{M}=\mathbf {b} _{1}\otimes \mathbf {b} _{2}\otimes \cdots \otimes \mathbf {b} _{M}}$ if and only if there exist ${\displaystyle \lambda _{k}}$ such that ${\displaystyle \lambda _{1}\lambda _{2}\cdots \lambda _{M}=1}$ and ${\displaystyle \mathbf {a} _{m}=\lambda _{m}\mathbf {b} _{m}}$ for all m. For this reason, the parameters ${\displaystyle \{\mathbf {a} _{m}\}_{m=1}^{M}}$ of a rank-1 tensor ${\displaystyle {\mathcal {A}}}$ are called identifiable or essentially unique. A rank-${\displaystyle r}$ tensor ${\displaystyle {\mathcal {A}}\in F^{I_{1}}\otimes F^{I_{2}}\otimes \cdots \otimes F^{I_{M}}}$ is called identifiable if every of its tensor rank decompositions is the sum of the same set of ${\displaystyle r}$ distinct tensors ${\displaystyle \{{\mathcal {A}}_{1},{\mathcal {A}}_{2},\ldots ,{\mathcal {A}}_{r}\}}$ where the ${\displaystyle {\mathcal {A}}_{i}}$'s are of rank 1. An identifiable rank-${\displaystyle r}$ thus has only one essentially unique decomposition

$${\displaystyle {\mathcal {A}}=\sum _{i=1}^{r}{\mathcal {A}}_{i},}$$

and all ${\displaystyle r!}$ tensor rank decompositions of ${\displaystyle {\mathcal {A}}}$ can be obtained by permuting the order of the summands. Observe that in a tensor rank decomposition all the ${\displaystyle {\mathcal {A}}_{i}}$'s are distinct, for otherwise the rank of ${\displaystyle {\mathcal {A}}}$ would be at most ${\displaystyle r-1}$.

Order-2 tensors in ${\displaystyle F^{I_{1}}\otimes F^{I_{2}}\simeq F^{I_{1}\times I_{2}}}$, i.e., matrices, are not identifiable for ${\displaystyle r>1}$. This follows essentially from the observation

$${\displaystyle {\mathcal {A}}=\sum _{i=1}^{r}\mathbf {a} _{i}\otimes \mathbf {b} _{i}=\sum _{i=1}^{r}\mathbf {a} _{i}\mathbf {b} _{i}^{T}=AB^{T}=(AX^{-1})(BX^{T})^{T}=\sum _{i=1}^{r}\mathbf {c} _{i}\mathbf {d} _{i}^{T}=\sum _{i=1}^{r}\mathbf {c} _{i}\otimes \mathbf {d} _{i},}$$

where ${\displaystyle X\in \mathrm {GL} _{r}(F)}$ is an invertible ${\displaystyle r\times r}$ matrix, ${\displaystyle A=[\mathbf {a} _{i}]_{i=1}^{r}}$ , ${\displaystyle B=[\mathbf {b} _{i}]_{i=1}^{r}}$, ${\displaystyle AX^{-1}=[\mathbf {c} _{i}]_{i=1}^{r}}$ and ${\displaystyle BX^{T}=[\mathbf {d} _{i}]_{i=1}^{r}}$. It can be shown[16] that for every ${\displaystyle X\in \mathrm {GL} _{n}(F)\setminus Z}$, where ${\displaystyle Z}$ is a closed set in the Zariski topology, the decomposition on the right-hand side is a sum of a different set of rank-1 tensors than the decomposition on the left-hand side, entailing that order-2 tensors of rank ${\displaystyle r>1}$ are generically not identifiable.

The situation changes completely for higher-order tensors in ${\displaystyle F^{I_{1}}\otimes F^{I_{2}}\otimes \cdots \otimes F^{I_{M}}}$ with ${\displaystyle M>2}$ and all ${\displaystyle I_{m}\geq 2}$. For simplicity in notation, assume without loss of generality that the factors are ordered such that ${\displaystyle I_{1}\geq I_{2}\geq \cdots \geq I_{M}\geq 2}$. Let ${\displaystyle S_{r}\subset F^{I_{1}}\otimes \cdots F^{I_{m}}\otimes \cdots \otimes F^{I_{M}}}$denote the set of tensors of rank bounded by ${\displaystyle r}$. Then, the following statement was proved to be correct using a computer-assisted proof for all spaces of dimension ${\displaystyle \Pi <15000}$,[17] and it is conjectured to be valid in general:[17][18][19]

There exists a closed set ${\displaystyle Z_{r}}$ in the Zariski topology such that every tensor ${\displaystyle {\mathcal {A}}\in S_{r}\setminus Z_{r}}$is identifiable (${\displaystyle S_{r}}$ is called generically identifiable in this case), unless either one of the following exceptional cases holds:

1. The rank is too large: ${\displaystyle r>r_{E}(I_{1},I_{2},\ldots ,I_{M})}$;
2. The space is identifiability-unbalanced, i.e., ${\textstyle I_{1}>\prod _{m=2}^{M}i_{m}-\sum _{m=2}^{M}(I_{m}-1)}$, and the rank is too large: ${\textstyle r\geq \prod _{m=2}^{M}I_{m}-\sum _{m=2}^{M}(I_{m}-1)}$;
3. The space is the defective case ${\displaystyle F^{4}\otimes F^{4}\otimes F^{3}}$ and the rank is ${\displaystyle r=5}$;
4. The space is the defective case ${\displaystyle F^{n}\otimes F^{n}\otimes F^{2}\otimes F^{2}}$, where ${\displaystyle n\geq 2}$, and the rank is ${\displaystyle r=2n-1}$;
5. The space is ${\displaystyle F^{4}\otimes F^{4}\otimes F^{4}}$ and the rank is ${\displaystyle r=6}$;
6. The space is ${\displaystyle F^{6}\otimes F^{6}\otimes F^{3}}$ and the rank is ${\displaystyle r=8}$; or
7. The space is ${\displaystyle F^{2}\otimes F^{2}\otimes F^{2}\otimes F^{2}\otimes F^{2}}$ and the rank is ${\displaystyle r=5}$.
8. The space is perfect, i.e., ${\textstyle r_{E}(I_{1},I_{2},\ldots ,I_{M})={\frac {\Pi }{\Sigma +1}}}$ is an integer, and the rank is ${\textstyle r=r_{E}(I_{1},I_{2},\ldots ,I_{M})}$.

In these exceptional cases, the generic (and also minimum) number of complex decompositions is

• proved to be ${\displaystyle \infty }$ in the first 4 cases;
• proved to be two in case 5;[20]
• expected[21] to be six in case 6;
• proved to be two in case 7;[22] and
• expected[21] to be at least two in case 8 with exception of the two identifiable cases ${\displaystyle F^{5}\otimes F^{4}\otimes F^{3}}$ and ${\displaystyle F^{3}\otimes F^{2}\otimes F^{2}\otimes F^{2}}$.

In summary, the generic tensor of order ${\displaystyle M>2}$ and rank ${\textstyle r<{\frac {\Pi }{\Sigma +1}}}$ that is not identifiability-unbalanced is expected to be identifiable (modulo the exceptional cases in small spaces).

The rank approximation problem asks for the rank-${\displaystyle r}$ decomposition closest (in the usual Euclidean topology) to some rank-${\displaystyle s}$ tensor ${\displaystyle {\mathcal {A}}}$, where ${\displaystyle r<s}$. That is, one seeks to solve

${\displaystyle \min _{\mathbf {a} _{i}^{m}\in F^{I_{m}}}\|{\mathcal {A}}-\sum _{i=1}^{r}\mathbf {a} _{i}^{1}\otimes \mathbf {a} _{i}^{2}\otimes \cdots \otimes \mathbf {a} _{i}^{M}\|_{F},}$

where ${\displaystyle \|\cdot \|_{F}}$ is the Frobenius norm.

It was shown in a 2008 paper by de Silva and Lim[8] that the above standard approximation problem may be ill-posed. A solution to aforementioned problem may sometimes not exist because the set over which one optimizes is not closed. As such, a minimizer may not exist, even though an infimum would exist. In particular, it is known that certain so-called border tensors may be approximated arbitrarily well by a sequence of tensor of rank at most ${\displaystyle r}$, even though the limit of the sequence converges to a tensor of rank strictly higher than ${\displaystyle r}$. The rank-3 tensor

${\displaystyle {\mathcal {A}}=\mathbf {u} \otimes \mathbf {u} \otimes \mathbf {v} +\mathbf {u} \otimes \mathbf {v} \otimes \mathbf {u} +\mathbf {v} \otimes \mathbf {u} \otimes \mathbf {u} ,\quad {\text{with }}\|\mathbf {u} \|=\|\mathbf {v} \|=1{\text{ and }}\langle \mathbf {u} ,\mathbf {v} \rangle \neq 1}$

can be approximated arbitrarily well by the following sequence of rank-2 tensors

${\displaystyle {\mathcal {A}}_{n}=n(\mathbf {u} +{\frac {1}{n}}\mathbf {v} )\otimes (\mathbf {u} +{\frac {1}{n}}\mathbf {v} )\otimes (\mathbf {u} +{\frac {1}{n}}\mathbf {v} )-n\mathbf {u} \otimes \mathbf {u} \otimes \mathbf {u} }$

as ${\displaystyle n\to \infty }$. This example neatly illustrates the general principle that a sequence of rank-${\displaystyle r}$ tensors that converges to a tensor of strictly higher rank needs to admit at least two individual rank-1 terms whose norms become unbounded. Stated formally, whenever a sequence

${\displaystyle {\mathcal {A}}_{n}=\sum _{i=1}^{r}\mathbf {a} _{i,n}^{1}\otimes \mathbf {a} _{i,n}^{2}\otimes \cdots \otimes \mathbf {a} _{i,n}^{M}}$

has the property that ${\displaystyle {\mathcal {A}}_{n}\to {\mathcal {A}}}$ (in the Euclidean topology) as ${\displaystyle n\to \infty }$, then there should exist at least ${\displaystyle 1\leq i\neq j\leq r}$ such that

${\displaystyle \|\mathbf {a} _{i,n}^{1}\otimes \mathbf {a} _{i,n}^{2}\otimes \cdots \otimes \mathbf {a} _{i,n}^{M}\|_{F}\to \infty {\text{ and }}\|\mathbf {a} _{j,n}^{1}\otimes \mathbf {a} _{j,n}^{2}\otimes \cdots \otimes \mathbf {a} _{j,n}^{M}\|_{F}\to \infty }$

as ${\displaystyle n\to \infty }$. This phenomenon is often encountered when attempting to approximate a tensor using numerical optimization algorithms. It is sometimes called the problem of diverging components. It was, in addition, shown that a random low-rank tensor over the reals may not admit a rank-2 approximation with positive probability, leading to the understanding that the ill-posedness problem is an important consideration when employing the tensor rank decomposition.

A common partial solution to the ill-posedness problem consists of imposing an additional inequality constraint that bounds the norm of the individual rank-1 terms by some constant. Other constraints that result in a closed set, and, thus, well-posed optimization problem, include imposing positivity or a bounded inner product strictly less than unity between the rank-1 terms appearing in the sought decomposition.