A tensor is a multidimensional or Nway array. A 1way tensor
is a vector and a 2way tensor is a matrix. A 3way tensor is a cube of
data. And so on.
A sparse tensor is a tensor where only a small fraction of the
elements are nonzero. In this case, it is more efficient to store just
the nonzeros and their indices.
A tensor that is decomposed as a Tucker Operator comprises a
core tensor multiplied in each mode by a matrix. For a threeway array,
this means the tensor X can be written as:
x_{ijk} = Σ_{r }Σ_{s} Σ_{t} g_{rst}
a_{ir} b_{js} c_{kt} for all i,j,k
Thus, the tensor X may be stored in terms of its components, the core
tensor G and the factor matrices A,B,C.
A tensor that is decomposed as a Kruskal Operator comprises a
component matrix for each mode and an optional scaling vector. For a
threeway array, this means the tensor X can be written as:
x_{ijk} = Σ_{r} λ_{r} a_{ir} b_{jr} c_{kr} for all i,j,k.
Thus, the tensor X may be stored in terms of its components, the
vector λ and the factor matrices A,B,C.
