A Note on Tensors Sample Clauses

A Note on Tensors. ‌ Readers familiar with tensors may recognize the Kronecker product as a tensor opera- tion and the Kronecker summation decomposition as a type of tensor decomposition. In this work, we focus on a linear operator (matrix-vector oriented) perspective of the Kronecker product rather than a tensor perspective. Yet, a great deal of work has been done on tensor SVDs. Here we detail some of the relevant work done from a tensor perspective, and explain how our work relates to the tensor-based approaches. Much of the discussion here stems from a report by Xxxxxx and Xxxxxx on a 2004 tensor workshop [31]. Their report uses notation and language that is accessible for those who are unfamiliar with tensor notation but are comfortable with linear al- gebraic formulations, and we suggest it as a starting point for those interested in learning more about tensor representations. Additionally, the survey by Grasedyck on low-rank tensor approximations [23] provides a more in-depth discussion of tensor representations. The SVD of a matrix is a form of tensor decomposition in which we express a matrix, which is a rank two tensor, as a sum of products of rank one tensors. The singular value matrix Σ is called a core tensor which determines how the singular vectors are combined to form the full matrix K [12, 31]. We will use × to denote a tensor product so as to differentiate it from our matrix-oriented Kronecker product; in literature, ◦ is often used as well [31]. Using the same notation for the singular value decomposition as before with K = U ΣV , we can write the tensor form of an SVD of a N × N matrix K as N N Σ × U × V T = Σ Σ σij(ui × vj) i=1 j=1 N N = Σ Σ σijuivT i=1 j=1 which, because Σ is diagonal, Σ= σ u vii i N T i Σ i=1 N = σii(ui × vi). Here, Σ is a core tensor: it dictates how the rank one basis tensors ui and vi are combined to form the matrix K [31]. Tensor decompositions, including the Xxxxxx decomposition and higher order SVD, often seek to represent a rank m tensor K in a similar form as above, but where tensor products of m rank 1 tensors are used in the decomposition. Let Ni denote the size of K along the ith index, then the decompositions are of the form X0 X0 Nm 2 K = X X . . . X xx ,x ,...,i i1 i1 im m i1 i2 im u(1) × u(2) × ... × u(m) = Σ × U (1) × U (2) × ... × U (m). Here, Σ is the core tensor with the same number of indices as K. The core dictates how the columns of the matrices U (i) are combined to form the tensor K. For general tensor decomposi...