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In machine learning, the term tensor informally refers to two different concepts (i) a way of organizing data and (ii) a multilinear (tensor) transformation. Data may be organized in a multidimensional array (M-way array), informally referred to as a "data tensor"; however, in the strict mathematical sense, a tensor is a multilinear mapping over a set of domain vector spaces to a range vector ...
[2] [3] Applying the canonical pairing to the kth V factor and the lth V ∗ factor, and using the identity on all other factors, defines the (k, l) contraction operation, which is a linear map that yields a tensor of type (m − 1, n − 1). [2] By analogy with the (1, 1) case, the general contraction operation is sometimes called the trace.
The total number of indices is also called the order, degree or rank of a tensor, [2] [3] [4] although the term "rank" generally has another meaning in the context of matrices and tensors. Just as the components of a vector change when we change the basis of the vector space, the components of a tensor also change under such a transformation.
Multilinear algebra is the study of functions with multiple vector-valued arguments, with the functions being linear maps with respect to each argument. It involves concepts such as matrices, tensors, multivectors, systems of linear equations, higher-dimensional spaces, determinants, inner and outer products, and dual spaces.
Xerus [52] is a C++ tensor algebra library for tensors of arbitrary dimensions and tensor decomposition into general tensor networks (focusing on matrix product states). It offers Einstein notation like syntax and optimizes the contraction order of any network of tensors at runtime so that dimensions need not be fixed at compile-time.
In mathematics, Voigt notation or Voigt form in multilinear algebra is a way to represent a symmetric tensor by reducing its order. [1] There are a few variants and associated names for this idea: Mandel notation, Mandel–Voigt notation and Nye notation are others found.
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As an example, a mixed tensor of type (1, 2) can be obtained by raising an index of a covariant tensor of type (0, 3), =, where is the same tensor as , because =, with Kronecker δ acting here like an identity matrix.