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On the other hand, a randomly sampled complex tensor of the same size will be a rank-1 tensor with probability zero, a rank-2 tensor with probability one, and a rank-3 tensor with probability zero. It is even known that the generic rank-3 real tensor in R 2 ⊗ R 2 ⊗ R 2 {\displaystyle \mathbb {R} ^{2}\otimes \mathbb {R} ^{2}\otimes \mathbb ...
In multilinear algebra, the higher-order singular value decomposition (HOSVD) of a tensor is a specific orthogonal Tucker decomposition.It may be regarded as one type of generalization of the matrix singular value decomposition.
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 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 products. In the (0, M ) -entry of the table, M denotes the dimensionality of the underlying vector space or manifold because for each dimension of the space, a separate index is needed to ...
Matrix rank should not be confused with tensor order, which is called tensor rank. Tensor order is the number of indices required to write a tensor , and thus matrices all have tensor order 2. More precisely, matrices are tensors of type (1,1), having one row index and one column index, also called covariant order 1 and contravariant order 1 ...
For example, the Ricci tensor is a non-metric contraction of the Riemann curvature tensor, and the scalar curvature is the unique metric contraction of the Ricci tensor. One can also view contraction of a tensor field in the context of modules over an appropriate ring of functions on the manifold [ 5 ] or the context of sheaves of modules over ...
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 (0,0) tensor is a number in the field . A (1,0) tensor is a vector. A (0,1) tensor is a covector. A (0,2) tensor is a bilinear form. An example is the metric tensor . A (1,1) tensor is a linear map.