Tensor reshaping

In multilinear algebra, a reshaping is any bijective map between an order-d tensor and an order-k tensor, where k < d.

Definition

Let V_k be an n_k-dimensional vector space over a field $F$ . Then there exist isomorphisms of vector spaces

{\begin{aligned}V_{1}\otimes \cdots \otimes V_{d}&\simeq F^{n_{1}}\otimes \cdots \otimes F^{n_{d}}\\&\simeq F^{n_{\pi _{1}}}\otimes \cdots \otimes F^{n_{\pi _{d}}}\\&\simeq F^{n_{\pi _{1}}n_{\pi _{2}}}\otimes F^{n_{\pi _{3}}}\otimes \cdots \otimes F^{n_{\pi _{d}}}\\&\simeq F^{n_{\pi _{1}}n_{\pi _{3}}}\otimes F^{n_{\pi _{2}}}\otimes F^{n_{\pi _{4}}}\otimes \cdots \otimes F^{n_{\pi _{d}}}\\&\,\,\,\vdots \\&\simeq F^{n_{1}n_{2}\ldots n_{d}},\end{aligned}}

where $\pi \in {\mathfrak {S}}_{d}$ is any permutation.

Via these (and other) isomorphisms, a tensor can be interpreted in several ways as an order-k tensor with $k<d$ . Let

{\begin{aligned}\Pi _{S}:V_{1}\otimes V_{2}\otimes \cdots \otimes \cdots \otimes V_{d}&\to V_{s_{1}}\otimes V_{s_{2}}\otimes \cdots \otimes V_{s_{|S|}}\\v_{1}\otimes v_{2}\otimes \cdots \otimes v_{d}&\mapsto v_{s_{1}}\otimes v_{s_{2}}\otimes \cdots \otimes v_{s_{|S|}}\end{aligned}}

denote the projection with $S=\{s_{1},s_{2},\ldots ,s_{k}\}\subset [1,d]$ . Let $S_{1}\sqcup S_{2}\sqcup \cdots \sqcup S_{\ell }=[1,d]$ be a partition. Then, the $(S_{1},S_{2},\ldots ,S_{\ell })$ -flattening of a simple tensor ${\mathcal {A}}=v_{1}\otimes v_{2}\otimes \cdots \otimes v_{d}$ is defined as

{\begin{aligned}{\mathcal {A}}_{(S_{1},S_{2},\ldots ,S_{\ell })}:={}&(\Pi _{S_{1}}{\mathcal {A}})\otimes (\Pi _{S_{2}}{\mathcal {A}})\otimes \cdots \otimes (\Pi _{S_{\ell }}{\mathcal {A}})\\[4pt]&\in \left(\bigotimes _{s\in S_{1}}V_{s}\right)\otimes \cdots \otimes \left(\bigotimes _{s\in S_{\ell }}V_{s}\right),\end{aligned}}

which may be interpreted as a tensor of order $\ell <d$ by ignoring the tensor product structure on $\otimes _{s\in S_{k}}V_{s}$ . The standard way to ignore the tensor product structure consists of looking at ${\mathcal {A}}_{(S_{1},\ldots ,S_{\ell })}$ through an isomorphism $\otimes _{s\in S_{k}}V_{s}\simeq F^{\prod _{s\in S_{k}}n_{s}}$ , in which case ${\mathcal {A}}_{(S_{1},\ldots ,S_{\ell })}$ is viewed as an element of $F^{N_{S_{1}}}\otimes \cdots \otimes F^{N_{S_{\ell }}}$ , where

${\textstyle N_{S_{k}}:=\prod _{s\in S_{k}}n_{s}}$ .

For general tensors ${\mathcal {A}}\in V_{1}\otimes V_{2}\otimes \cdots \otimes V_{d}$ the above definition is extended linearly as follows. Since every tensor admits a tensor rank decomposition, we can define the $(S_{1},S_{2},\ldots ,S_{\ell })$ -flattening of ${\textstyle {\mathcal {A}}=\sum _{i=1}^{r}\mathbf {a} _{i}^{1}\otimes \mathbf {a} _{i}^{2}\otimes \cdots \otimes \mathbf {a} _{i}^{d}}$

as follows:

{\mathcal {A}}_{(S_{1},S_{2},\ldots ,S_{\ell })}:=\sum _{i=1}^{r}\left(\mathbf {a} _{i}^{1}\otimes \mathbf {a} _{i}^{2}\otimes \cdots \otimes \mathbf {a} _{i}^{d}\right)_{(S_{1},S_{2},\ldots ,S_{\ell })}.

Example

For example, assume that a basis of $V_{k}$ is $\{v_{1}^{k},v_{2}^{k},\ldots ,v_{n_{k}}^{k}\}$ and that the tensor is expressed with respect to this basis as

{\mathcal {A}}=\sum _{j_{1},j_{2},\ldots ,j_{d}}a_{j_{1},j_{2},\ldots ,j_{d}}v_{j_{1}}^{1}\otimes v_{j_{2}}^{2}\otimes \cdots \otimes v_{j_{d}}^{d}.

Via the first isomorphism, we can represent ${\mathcal {A}}$ as

{\mathcal {A}}=\sum _{j_{1},j_{2},\ldots ,j_{d}}a_{j_{1},j_{2},\ldots ,j_{d}}\mathbf {e} _{j_{1}}^{1}\otimes \mathbf {e} _{j_{2}}^{2}\otimes \cdots \otimes \mathbf {e} _{j_{d}}^{d},

where $\mathbf {e} _{j}^{k}$ is the jth standard basis vector of $F^{n_{k}}$ . Taking $S_{1}=[1,d]$ and a bijective map $\mu :[1,n_{1}]\times \cdots \times [1,n_{d}]\to [1,n_{1}\cdots n_{d}]$ , we can construct an isomorphism between $F^{n_{1}}\otimes \cdots \otimes F^{n_{d}}$ and $F^{n_{1}\cdots n_{d}}$ by mapping $\mathbf {e} _{j_{1}}^{1}\otimes \cdots \otimes \mathbf {e} _{j_{d}}^{d}\mapsto \mathbf {e} _{\mu (j_{1},j_{2},\ldots ,j_{d})},$ where $\mathbf {e} _{j}$ is the jth standard basis vector of $F^{n_{1}\cdots n_{d}}$ . Via this identification, we have $A_{(S_{1})}=[a_{\mu ^{-1}(i)}]_{i=1}^{n_{1}\cdots n_{d}}$ .

Vectorization

Let ${\mathcal {A}}\in F^{n_{1}\times n_{2}\times \cdots \times n_{d}}$ be the coordinate representation of an abstract tensor with respect to a basis. A vectorization of ${\mathcal {A}}$ is an $(S_{1})$ -reshaping wherein $S_{1}=[1,d]$ , so that the tensor is simply interpreted as a vector in $F^{n_{1}\cdots n_{d}}$ . A standard choice of $\mu$ is such that

\operatorname {vec} ({\mathcal {A}}):={\mathcal {A}}_{(S_{1})}={\begin{bmatrix}a_{1,1,\ldots ,1}&a_{2,1,\ldots ,1}&\cdots &a_{n_{1},1,\ldots ,1}&a_{1,2,1,\ldots ,1}&\cdots &a_{n_{1},n_{2},\ldots ,n_{d}}\end{bmatrix}}^{T},

which is consistent with the way in which the colon operator in Matlab and GNU Octave reshapes a higher-order tensor into a vector.

Flattenings

Let ${\mathcal {A}}\in F^{n_{1}\times n_{2}\times \cdots \times n_{d}}$ be the coordinate representation of an abstract tensor with respect to a basis. A standard factor-k flattening of ${\mathcal {A}}$ is an $(S_{1},S_{2})$ -reshaping in which $S_{1}=\{k\}$ and $S_{2}=[1,d]\smallsetminus S_{1}$ . Usually, a standard flattening is denoted by

{\mathcal {A}}_{(k)}:={\mathcal {A}}_{(S_{1},S_{2})}

These reshapings are sometimes called matricizations or unfoldings in the literature. A standard choice of $\mu$ is the one that is consistent with the reshape function in Matlab and GNU Octave, namely

{\mathcal {A}}_{(k)}:={\begin{bmatrix}a_{1,1,\ldots ,1,1,1,\ldots ,1}&a_{2,1,\ldots ,1,1,1,\ldots ,1}&\cdots &a_{n_{1},n_{2},\ldots ,n_{k-1},1,n_{k+1},\ldots ,n_{d}}\\a_{1,1,\ldots ,1,2,1,\ldots ,1}&a_{2,1,\ldots ,1,2,1,\ldots ,1}&\cdots &a_{n_{1},n_{2},\ldots ,n_{k-1},2,n_{k+1},\ldots ,n_{d}}\\\vdots &\vdots &&\vdots \\a_{1,1,\ldots ,1,n_{k},1,\ldots ,1}&a_{2,1,\ldots ,1,n_{k},1,\ldots ,1}&\cdots &a_{n_{1},n_{2},\ldots ,n_{k-1},n_{k},n_{k+1},\ldots ,n_{d}}\end{bmatrix}}

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