The strain and stress concentration factor tensors that relate the local average strain and stress fields, respectively, to the corresponding global average fields are derived according to the theory of average fields.
To overcome the weakness of the vector model, Grindlay and Vasilescu modeled HRTFs using a multi-factor (tensor) representation [21].
I will rephrase Strassen's conjecture in terms of tensor rank and show that it holds for three-factor tensors, whenever the dimension of one of the involved vector spaces is at most two.
Zhang, X; Yuan, X; Carin, L, Nonlocal Low-Rank Tensor Factor Analysis for Image Restoration, Proceedings of the Ieee Computer Society Conference on Computer Vision and Pattern Recognition (2018), pp. 8232-8241 [abs1109/CVPR.2018.00859] [abs].
Where χ P (a ⊗ ϕ ⊗ b ) = (− 1 ) | ϕ | | a | ϕ (a ⊗ u B A s (1 ) ⊗ b ) and υ P is the strict map sending p to the morphism ϕ p defined as ϕ p (a ⊗ q ⊗ b ) = μ P (a ⊗ q + ⊗ p ⊗ q − ⊗ b ) where q ± is the tensor factor of q in D s ±.
It is easy to see that Ψ factors through this tensor decomposition of U s 2, so we may cancel the U s 2 ′ factor and show the equivalence on the U s 0 factor.
All of the algorithms can be formulated as message passing algorithm on a graph where vertices correspond to indices and cliques represent factors of the tensor decomposition.
Note that (R_1) and (R_0) commute, as they act on different factors of the tensor product (fancyscript {A}_V).
MPCA is a multilinear extension of Principal Component Analysis (PCA) [5] that analyzes the interaction between multiple factors utilizing a tensor framework.
For applications, where the multi-dimensional data obeys a PARAFAC decomposition, it is important to estimate the factors of the tensor, which are defined as, and we assume that the rank of each F r)is equal to min(M r, d).
