Tensor reshaping

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In multilinear algebra, a reshaping of tensors is any bijection between the set of indices of an order- tensor and the set of indices of an order- tensor, where . The use of indices presupposes tensors in coordinate representation with respect to a basis. The coordinate representation of a tensor can be regarded as a multi-dimensional array, and a bijection from one set of indices to another therefore amounts to a rearrangement of the array elements into an array of a different shape. Such a rearrangement constitutes a particular kind of linear map between the vector space of order- tensors and the vector space of order- tensors.

Definition[edit]

Given a positive integer , the notation refers to the set of the first d positive integers.

For each integer where for a positive integer , let Vk denote an nk-dimensional vector space over a field . Then there are vector space isomorphisms (linear maps)

where is any permutation and is the symmetric group on elements. Via these (and other) vector space isomorphisms, a tensor can be interpreted in several ways as an order- tensor where .

Coordinate representation[edit]

The first vector space isomorphism on the list above, , gives the coordinate representation of an abstract tensor. Assume that each of the vector spaces has a basis . The expression of a tensor with respect to this basis has the form

where the coefficients are elements of . The coordinate representation of is
where is the standard basis vector of . This can be regarded as a d-dimensional array whose elements are the coefficients .

Vectorization[edit]

By means of a bijective map , a vector space isomorphism between and is constructed via the mapping where for every natural number such that , the vector denotes the jth standard basis vector of . In such a reshaping, the tensor is simply interpreted as a vector in . This is known as vectorization, and is analogous to vectorization of matrices. A standard choice of bijection is such that

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

General flattenings[edit]

For any permutation there is a canonical isomorphism between the two tensor products of vector spaces and . Parentheses are usually omitted from such products due to the natural isomorphism between and , but may, of course, be reintroduced to emphasize a particular grouping of factors. In the grouping,

there are groups with factors in the group (where and ).

Letting for each satisfying , an -flattening of a tensor , denoted , is obtained by applying the two processes above within each of the groups of factors. That is, the coordinate representation of the group of factors is obtained using the isomorphism , which requires specifying bases for all of the vector spaces . The result is then vectorized using a bijection to obtain an element of , where , the product of the dimensions of the vector spaces in the group of factors. The result of applying these isomorphisms within each group of factors is an element of , which is a tensor of order .

The vectorization of is an -reshaping, wherein .

Matricization[edit]

Let be the coordinate representation of an abstract tensor with respect to a basis. A standard factor-k flattening of is an -reshaping in which and . Usually, a standard flattening is denoted by

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