Symmetric Tensor Decompositions On Varieties

This paper focuses on computing symmetric tensor decompositions on a given set $X$. We assume that $X\subseteq\mathbb{C}^{n+1}$ is a variety that is given by homogeneous polynomial equations. Generally, symmetric tensor decompositions on $X$ can be theoretically studied by secant varieties of the Veronese embedding of $X$ [28, 42, 43]. From this view, one may expect to get polynomials which characterize the symmetric $X$-rank of a given symmetric tensor. In this paper, we give a method for computing symmetric $X$-rank decompositions. It is based on the tool of generating polynomials that were recently introduced in [33]. The work [33] only discussed the case $X=\mathbb{C}^{n+1}$. When $X$ is a variety, i.e., the method in [33] does not work, because $u_{i}\in X$ is required in (1.1). We need to modify the approach in [33], by posing additional conditions on generating polynomials. For this purpose, we give a unified framework for computing symmetric tensor decompositions on $X$, which sheds light on the study of both theoretical and computational aspects of tensor decompositions.

The paper is organized as follows. Section 2 gives some basics for tensor decompositions. Section 3 studies some properties of symmetric tensor decompositons on $X$. Section 4 defines generating polynomials and generating matrices. It gives conditions ensuring that the computed vectors belong to the given set in symmetric tensor decompositions. Section 5 presents a unified framework to do the computation. Last, Section 6 gives various examples to show how the proposed method works.

There is a one-to-one correspondence between symmetric tensors in $\operatorname{S}^{d}(\mathbb{C}^{n+1})$ and homogeneous polynomials of degree $d$ and in $(n+1)$ variables. When $\mathcal{A}\in\operatorname{S}^{d}(\mathbb{C}^{n+1})$ is symmetric, we can equivalently use $\alpha=(\alpha_{1},\dots,\alpha_{n})\in\mathbb{N}_{d}^{n}$ to label $\mathcal{A}$ in the way that

The symmetry guarantees that the labelling $\mathcal{A}_{\alpha}$ is well-defined. For $\mathcal{A}$, define the homogeneous polynomial (i.e., a form) in $x:=(x_{0},\dots,x_{n})$ and of degree $d$:

If $\mathcal{A}$ is labeled as in (2.2), then

where $\binom{d}{\alpha_{0},\dots,\alpha_{n}}\coloneqq\frac{d!}{\alpha_{0}!\cdots\alpha_{n}!}.$ The decomposition (1.1) is equivalent to

For a polynomial $p=\sum_{\alpha\in\mathbb{N}^{n}_{d}}p_{\alpha}y_{1}^{\alpha_{1}}\cdots y_{n}^{\alpha_{n}}$ and $\mathcal{A}\in\operatorname{S}^{d}(\mathbb{C}^{n+1})$, define the operation

where $\mathcal{A}$ is labeled as in (2.2). For fixed $p$, $\langle p,\cdot\rangle$ is a linear functional on $\operatorname{S}^{d}(\mathbb{C}^{n+1})$, while for fixed $\mathcal{A}\in\operatorname{S}^{d}(\mathbb{C}^{n+1})$, $\langle\cdot,\mathcal{A}\rangle$ is a linear functional on $\mathbb{C}[y_{1},\ldots,y_{n}]_{d}$.

A set $I\subseteq\mathbb{C}[x]:=\mathbb{C}[x_{0},\dots,x_{n}]$ is an ideal if $I\cdot\mathbb{C}[x]\subseteq I$ and $I+I\subseteq I$. For polynomials $f_{1},\dots,f_{s}\in\mathbb{C}[x]$, let $\left\langle f_{1},\dots,f_{s}\right\rangle$ denote the smallest ideal that contains $f_{1},\ldots,f_{s}$. A subset $X\subseteq\mathbb{C}^{n+1}$ is called an affine variety if $X$ is the set of common zeros of some polynomials in $\mathbb{C}[x]$. The vanishing ideal of $X$ is the ideal consisting of all polynomials identically vanish on $X$.

Two nonzero vectors in $\mathbb{C}^{n+1}$ are equivalent if they are parallel to each other. Denote by $[u]$ the set of all nonzero vectors that are equivalent to $u$; the set $[u]$ is called the equivalent class of $u$. The set of all equivalent classes $[u]$ with $0\neq u\in\mathbb{C}^{n+1}$ is the projective space $\mathbb{P}^{n}$, or equivalently, $\mathbb{P}^{n}=\{[u]:\,0\neq u\in\mathbb{C}^{n+1}\}$. A subset $Z\subseteq\mathbb{P}^{n}$ is said to be a projective variety if there are homogeneous polynomials $h_{1},\ldots,h_{t}\in\mathbb{C}[x]$ such that

The vanishing ideal $\mathcal{I}(Z)$ is defined to be the ideal consisting of all polynomials identically vanish on $Z$. For each nonnegative integer $m$, we denote by $\mathcal{I}_{m}(Z)$ the linear subspace of polynomials of degree $m$ in $\mathcal{I}(Z)$. A projective variety $Z\subseteq\mathbb{P}^{n}$ is said to be nondegenerate if $Z$ is not contained in any proper linear subspace of $\mathbb{P}^{n}$, i.e., $\mathcal{I}_{1}(Z)=\{0\}$.

In the Zariski topology for $\mathbb{C}^{n+1}$ and $\mathbb{P}^{n}$, the closed sets are varieties and the open sets are complements of varieties. For a projective variety $Z\subseteq\mathbb{P}^{n}$, its Hilbert function $h_{Z}:\mathbb{N}\to\mathbb{N}$ is defined as $h_{Z}(d)\,\coloneqq\,\dim\mathbb{C}[Z]_{d}.$ As in [11, 20, 21], when $d$ is sufficiently large, $h_{Z}(d)$ is a polynomial and

where $O(d^{m-1})$ denotes terms of order at most $m-1$.

For an affine variety $X\subseteq\mathbb{C}^{n+1}$, we denote by $\mathbb{P}X$ the projective set of equivalent classes of nonzero vectors in $X$, i.e., $\mathbb{P}X=\{[u]:\,0\neq u\in X\}.$ If $\mathbb{P}X=Z$, then $X$ is called the affine cone of $Z$. Clearly, the vanishing ideal of $X$ is the same as that of $\mathbb{P}X$, i.e., $\mathcal{I}(X)=\mathcal{I}(\mathbb{P}X)$. For a degree $m>0$, the set

is the space of all forms of degree $m$ vanishing on $X$.

For an affine variety $X\subseteq\mathbb{C}^{n+1}$, let $v_{d}(X)\subseteq\operatorname{S}^{d}(\mathbb{C}^{n+1})$ be the image of $X$ under the Veronese map:

Note that $v_{d}^{-1}(u^{\otimes d})=\{\omega^{i}u:i=0,\dots,d-1\}$, where $\omega$ is a primitive $d$-th root of $1$. Therefore, the dimension of $v_{d}(X)$ is the same as that of $X$. In particular, for

the set $v_{d}(C)$ is a variety of tensors $\mathcal{A}\in\operatorname{S}^{d}(\mathbb{C}^{2})$ that is defined by the equations

In the above, the tensors in $\operatorname{S}^{d}(\mathbb{C}^{2})$ are labelled by vectors in $\mathbb{N}_{d}^{2}$. The projectivization $\mathbb{P}v_{d}(C)$ is called the rational normal curve in the projective space $\mathbb{P}\operatorname{S}^{d}(\mathbb{C}^{2})\simeq\mathbb{P}^{d}$. For a projective variety $Z\subseteq\mathbb{P}^{n}$, the Veronese embedding map $v_{d}$ is defined in the same way as

Note that $\mathbb{P}v_{d}(X)=v_{d}(\mathbb{P}X)$ for every affine variety $X$.

For projective spaces $\mathbb{P}^{n_{1}},\ldots,\mathbb{P}^{n_{k}}$, their Segre product, denoted as $\operatorname{Seg}(\mathbb{P}^{n_{1}}\times\cdots\times\mathbb{P}^{n_{k}})$, is the image of the Segre map:

The dimension of $\operatorname{Seg}(\mathbb{P}^{n_{1}}\times\cdots\times\mathbb{P}^{n_{k}})$ is the sum $n_{1}+\cdots+n_{k}$. The Segre product $\mathbb{P}^{n_{1}},\dots,\mathbb{P}^{n_{k}}$ is defined by equations of the form

Here, tensors in $\mathbb{P}(\bigotimes_{j=1}^{k}\mathbb{C}^{n_{j}+1})$ are labelled by integral tuples in $\prod_{j=1}^{k}\{0,\dots,n_{j}\}$.

Let $X\subseteq\mathbb{C}^{n+1}$ be an affine variety and let $v_{d}(X)$ be its image under the $d$-th Veronese map $v_{d}$. Define the set

The Zariski closure of $\sigma_{r}^{\circ}(v_{d}(X))$, which we denote as $\sigma_{r}(v_{d}(X))\subseteq\mathbb{C}^{n+1}$, is called the $r$th secant variety of $v_{d}(X)$. The closure $\sigma_{r}(v_{d}(X))$ is an affine variety in $\operatorname{S}^{d}(\mathbb{C}^{n+1})$, while $\sigma_{r}^{\circ}(v_{d}(X))$ is usually not. However, it holds that

because $\sigma^{\circ}_{r}(v_{d}(X))$ is a dense subset of $\sigma_{r}(v_{d}(X))$ in the Zariski topology. When $v_{d}(X)$ is replaced by a general variety $Y$, the sets $\sigma_{r}^{\circ}(Y)$ and $\sigma_{r}(Y)$ can be defined in the same way. We refer to [27] for secant varieties.

Let $\mathbb{P}X\subseteq\mathbb{P}^{n}$ be the projectivization of $X$. Its vanishing ideal is $\mathcal{I}(X)$, the ideal of polynomials that are identically zero on $X$. The section of degree-$d$ forms in $\mathcal{I}(X)$ is

Let $c:=\dim\mathcal{I}_{d}(X)$. Choose a basis $\{f_{1},\ldots,f_{c}\}$ for $\mathcal{I}_{d}(X)$. Let $l_{1},\dots,l_{c}$ be linear functionals on $\operatorname{S}^{d}(\mathbb{C}^{n+1})$ such that

They are linearly independent functions on $\operatorname{S}^{d}(X)$. If we use $\hat{f}_{i}$ to denote the dehomogenization of $f_{i}$, i.e., $\hat{f}_{i}(x_{1},\ldots,x_{n}):=f_{i}(1,x_{1},\ldots,x_{n})$, then $l_{i}(\mathcal{A})=\langle\hat{f}_{i},\mathcal{A}\rangle$ for all $\mathcal{A}\in\operatorname{S}^{d}(\mathbb{C}^{n+1})$. See (2.4) for the definition of the operation $\langle\cdot,\cdot\rangle$.

The first part of Proposition 3.1 is a high dimensional analogue of the apolarity lemma, which can be found in [23, Theorem 5.3], [41, Section 1.3], and [48, Section 3]). For convenience of referencing, we state this result here and give a straightforward proof. We would like to thank Zach Teitler for pointing out the relationship between Proposition 3.1 and the apolarity Lemma.

By Proposition 3.1, we get the following algorithm for checking if $\mathcal{A}\in\operatorname{S}^{d}(X)$ or not. Suppose the vanishing ideal $\mathcal{I}(X)$ is generated by the forms $f_{1},\dots,f_{s}$, with degrees $d_{1}\leq\dots\leq d_{s}$ respectively.

The above algorithm can be easily applied to detect tensors in $\operatorname{S}^{d}(X)$. For instance, if $X$ is defined by linear equations $\sum_{j=0}^{n}f_{ij}x_{j}=0$ ($i=1,\ldots,c$), then $\mathcal{A}\in\operatorname{S}^{d}(X)$ if and only if for all $1\leq i\leq c$ and $0\leq k_{2},\ldots,k_{m}\leq n$,

If $X$ is a hypersurface defined by a single homogeneous polynomial $f(x)=0$ with $\deg(f)\leq m$, then $\mathcal{A}\in\operatorname{S}^{d}(X)$ if and only if

for all monomials $x^{\alpha}$ with $\deg(f)+|\alpha|=m$.

When does $\operatorname{S}^{d}(X)=\operatorname{S}^{d}(\mathbb{C}^{n+1})$, i.e., when does every tensor admit a symmetric $X$-decomposition? By Proposition 3.1, we get the following characterization.

The above corollary implies that if $X$ is a hypersurface of degree bigger than $d$, then every tensor in $\operatorname{S}^{d}(\mathbb{C}^{n+1})$ has a symmetric $X$-decomposition. Moreover, if $X=\mathbb{C}^{n+1}$, then obviously we have $\mathcal{I}_{d}(X)=\{0\}$ for any $d\geq 1$, which implies the well known fact [9] that every symmetric tensor admits a symmetric decomposition.

By Proposition 3.1, $\dim\operatorname{S}^{d}(X)=h_{\mathbb{P}X}(d),$ where $h_{\mathbb{P}X}(\cdot)$ is the Hilbert function for the projective variety $\mathbb{P}X$. For the Veronese map $v_{d}$, we have $\dim v_{d}(X)=\dim X$. Therefore, the expected dimension of the secant variety $\sigma_{r}(v_{d}(X))$:

The expected generic symmetric $X$-rank of $\operatorname{S}^{d}(X)$ is therefore

When $\mathbb{P}X$ is a curve, we can get the dimension of $\sigma_{r}(v_{d}(X))$ as follows.

By [20, Example 13.7], if $\mathbb{P}X$ is a space curve of degree $e$, i.e., $\mathbb{P}X$ is a curve in $\mathbb{P}^{3}$ which intersects a generic plane in $e$ points, then $d_{0}=e-2$ in Proposition 3.5. For an arbitrary curve $\mathbb{P}X$, however, not much is known about $d_{0}$. The Hilbert functions are also known for Veronese varieties, Grassmann varieties and flag varieties, see [20, 19]. Hence the expected value of the generic rank for them may be calculated by (3.5).

A nonsymmetric tensor $\mathcal{A}\in(\mathbb{C}^{d+1})^{\otimes k}$ is said to admit a Vandermonde decomposition if there exist vectors ($s=1,\ldots,k$, $j=1,\ldots,r$)

such that

The smallest integer $r$ in the above is called the Vandermonde rank of $\mathcal{A}$, which we denote as $\operatorname{vrank}(\mathcal{A})$. Since $v_{s}^{(j)}=(a_{sj},b_{sj})^{\otimes d}\in\operatorname{S}^{d}\mathbb{C}^{2}\simeq\mathbb{C}^{d+1},$ we can rewrite (3.6) equivalently as

The Vandermonde decomposition can be thought of as a symmetric tensor decomposition on the set $X\subseteq\mathbb{C}^{2^{k}}$ such that

The variety $\sigma_{r}(v_{d}(X))\subseteq\mathbb{P}\operatorname{S}^{d}(\mathbb{C}^{2^{k}})$ is exactly the Zariski closure of tensors whose Vandermonde ranks at most $r$. The vanishing ideal of the Segre variety $\mathbb{P}^{n_{1}}\times\cdots\times\mathbb{P}^{n_{k}}$ is generated by $2\times 2$ minors of its flattenings [18]. So $\mathcal{I}_{d}(\mathbb{P}^{n_{1}}\times\cdots\times\mathbb{P}^{n_{k}})\neq 0$ for all $d\geq 2$. By Corollary 3.3, $\operatorname{S}^{d}(X)$ is a proper subspace of $\operatorname{S}^{d}(\mathbb{C}^{2^{k}})$. However, every $\mathcal{A}\in(\mathbb{C}^{d+1})^{\otimes k}$ has a Vandermonde decomposition.