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.
Given a positive integer
, the notation
refers to the set
of the first M positive integers.
For each integer
where
for a positive integer
, let
denote an
-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 M-way array whose elements are the coefficients
.
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
.
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 ith 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
.
The vectorization of
denoted with
or
is an
-reshaping where
and
.
Mode-m Flattening / Mode-m Matrixization
[edit]
Let
be the coordinate representation of an abstract tensor with respect to a basis.
Mode-m matrixizing (a.k.a. flattening) of
is an
-reshaping in which
and
. Usually, a standard matrixizing is denoted by
This reshaping is sometimes called matrixizing, matricizing, flattening or unfolding 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
Definition Mode-m Matrixizing:[1]
The mode-m matrixizing of a tensor
is defined as the matrix
. As the parenthetical ordering indicates, the mode-m column vectors are arranged by
sweeping all the other mode indices through their ranges,
with smaller mode indexes varying more rapidly than larger ones; thus