A tensor is a multidimensional or N-way array. A 1-way tensor is a vector and a 2-way tensor is a matrix. A 3-way 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 three-way 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 three-way 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.