multilinear algebra, the tensor rank decomposition or rank-R decomposition is the decomposition of a tensor as a sum of R rank-1 tensors, where R is minimal...

states, and operators or tensor trains; Online Tensor Decompositions hierarchical Tucker decomposition; block term decomposition This section introduces...

tensor"), may be analyzed either by artificial neural networks or tensor methods. Tensor decomposition factorizes data tensors into smaller tensors....

polyadic decomposition (CPD), which is a variant of the tensor rank decomposition, in which a tensor is approximated as a sum of K rank-1 tensors for a user-specified...

metric tensor, and is usually denoted g. A simple tensor (also called a tensor of rank one, elementary tensor or decomposable tensor) is a tensor that can...

In mathematics, a symmetric tensor is an unmixed tensor that is invariant under a permutation of its vector arguments: T ( v 1 , v 2 , … , v r ) = T (...

Tensor Processing Unit (TPU) Tensor rank decomposition Einstein Notation Spin network Orús, Román (5 August 2019). "Tensor networks for complex quantum...

unreliable, and a rank-revealing decomposition should be used instead. An effective alternative is the singular value decomposition (SVD), but there are...

the Schmidt decomposition (named after its originator Erhard Schmidt) refers to a particular way of expressing a vector in the tensor product of two...

elasticity tensor is a fourth-rank tensor describing the stress-strain relation in a linear elastic material. Other names are elastic modulus tensor and stiffness...

(electromagnetic tensor, Maxwell tensor, permittivity, magnetic susceptibility, ...), and general relativity (stress–energy tensor, curvature tensor, ...). In...

{\displaystyle L} . Tensor-CURT decomposition is a generalization of matrix-CUR decomposition. Formally, a CURT tensor approximation of a tensor A is three matrices...

m\times n} ⁠ matrix. It is related to the polar decomposition. Specifically, the singular value decomposition of an m × n {\displaystyle m\times n} complex...

called the tensor product of v {\displaystyle v} and w {\displaystyle w} . An element of V ⊗ W {\displaystyle V\otimes W} is a tensor, and the tensor product...

generalized singular value decomposition (GSVD) is the name of two different techniques based on the singular value decomposition (SVD). The two versions...

alternating tensor subspace. On the other hand, the image A ( T ( V ) ) {\displaystyle {\mathcal {A}}(\mathrm {T} (V))} is always the alternating tensor graded...

In multilinear algebra, a tensor contraction is an operation on a tensor that arises from the canonical pairing of a vector space and its dual. In components...

Hankel matrix (0,1)-matrix Matrix decomposition Cholesky decomposition LU decomposition QR decomposition Polar decomposition Reducing subspace Spectral theorem...

tensor is antisymmetric with respect to its first three indices. If a tensor changes sign under exchange of each pair of its indices, then the tensor...

algebra Multilinear Principal Component Analysis Tensor Tensor decomposition Tensor software Tucker decomposition M. A. O. Vasilescu, D. Terzopoulos (2003) "Multilinear...

Scalar–vector–tensor decomposition Hodge theory generalizing Helmholtz decomposition Polar factorization theorem Helmholtz–Leray decomposition used for defining...

differential geometry, a tensor density or relative tensor is a generalization of the tensor field concept. A tensor density transforms as a tensor field when passing...

decomposition, and mixed-canonical decomposition. The decomposition of the d N {\displaystyle d^{N}} -dimensional tensor starts with the separation of the...

notation and manipulation for tensors and tensor fields on a differentiable manifold, with or without a metric tensor or connection. It is also the modern...

Cauchy stress tensor (symbol ⁠ σ {\displaystyle {\boldsymbol {\sigma }}} ⁠, named after Augustin-Louis Cauchy), also called true stress tensor or simply stress...

structure tensor is often used in image processing and computer vision. For a function I {\displaystyle I} of two variables p = (x, y), the structure tensor is...

the Weyl tensor.) As one might expect from the Ricci decomposition of the Riemann tensor into the Weyl tensor plus a sum of fourth-rank tensors constructed...

{\mathcal {A}}} 's tensor rank decompositions. The validity of the above expression is not limited to a tensor rank decomposition; in fact, it is valid...

Basser PJ, Pajevic S (2007). "Spectral decomposition of a 4th-order covariance tensor: applications to diffusion tensor MRI". Signal Processing. 87 (2): 220–236...

He is best known for the tensor train decomposition, which is more commonly called a matrix product state in the area of tensor networks. Oseledets joined...