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The rank of a tensor depends on the field over which the tensor is decomposed. It is known that some real tensors may admit a complex decomposition whose rank is strictly less than the rank of a real decomposition of the same tensor. As an example, [8] consider the following real tensor
Some aspects can be traced as far back as F. L. Hitchcock in 1928, [1] but it was L. R. Tucker who developed for third-order tensors the general Tucker decomposition in the 1960s, [2] [3] [4] further advocated by L. De Lathauwer et al. [5] in their Multilinear SVD work that employs the power method, or advocated by Vasilescu and Terzopoulos ...
Thinking of matrices as tensors, the tensor rank generalizes to arbitrary tensors; for tensors of order greater than 2 (matrices are order 2 tensors), rank is very hard to compute, unlike for matrices. There is a notion of rank for smooth maps between smooth manifolds. It is equal to the linear rank of the derivative.
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 ...
A tensor whose components in an orthonormal basis are given by the Levi-Civita symbol (a tensor of covariant rank n) is sometimes called a permutation tensor. Under the ordinary transformation rules for tensors the Levi-Civita symbol is unchanged under pure rotations, consistent with that it is (by definition) the same in all coordinate systems ...
For a 3rd-order tensor , where is either or , Tucker Decomposition can be denoted as follows, = () where is the core tensor, a 3rd-order tensor that contains the 1-mode, 2-mode and 3-mode singular values of , which are defined as the Frobenius norm of the 1-mode, 2-mode and 3-mode slices of tensor respectively.
2.3 General rank. 3 See also. ... (1,1) tensor is a linear map. An example is the delta, ... Code of Conduct; Developers; Statistics; Cookie statement;