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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
2.3 General rank. 3 See also. ... Download QR code; Print/export Download as PDF; ... (1,1) tensor is a linear map. An example is the delta, ...
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 ...
The tensors are classified according to their type (n, m), where n is the number of contravariant indices, m is the number of covariant indices, and n + m gives the total order of the tensor. For example, a bilinear form is the same thing as a (0, 2)-tensor; an inner product is an example of a (0, 2)-tensor, but not all (0, 2)-tensors are inner ...
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.
For example, in the above example the null space is spanned by the last row of and the range is spanned by the first three columns of . As a consequence, the rank of M {\displaystyle \mathbf {M} } equals the number of non-zero singular values which is the same as the number of non-zero diagonal elements in Σ ...
Tensor-CURT decomposition [6] is a generalization of matrix-CUR decomposition. Formally, a CURT tensor approximation of a tensor A is three matrices and a (core-)tensor C, R, T and U such that C is made from columns of A, R is made from rows of A, T is made from tubes of A and that the product U(C,R,T) (where the ,,-th entry of it is ′, ′, ′ ′, ′, ′, ′, ′, ′) closely ...
A multi-way graph with K perspectives is a collection of K matrices ,..... with dimensions I × J (where I, J are the number of nodes). This collection of matrices is naturally represented as a tensor X of size I × J × K. In order to avoid overloading the term “dimension”, we call an I × J × K tensor a three “mode” tensor, where “modes” are the numbers of indices used to index ...