Gaussian Mixtures and Tensor Decompositions

1 Tensors Tensors are the generalizations of vectors v ∈ R and matrices M ∈ Rm×n. These are respectively 1-tensors and 2-tensors, which we can represent using oneand two-dimensional arrays of real numbers. Although there are more general definitions, for our purposes, a p-th order tensor (or a p-tensor) is an object that can be represented using a p-dimensional array of real numbers. Technically, today by “tensor” we’re strictly referring to covariant Cartesian tensors. We can add tensors of the same order and shape, and multiply them by scalars.