Decompositions of a Higher-Order Tensor in Block Terms—Part II: Definitions and Uniqueness

In this paper we introduce a new class of tensor decompositions. Intuitively, we decompose a given tensor block into blocks of smaller size, where the size is characterized by a set of mode-$n$ ranks. We study different types of such decompositions. For each type we derive conditions under which essential uniqueness is guaranteed. The parallel factor decomposition and Tucker's decomposition can be considered as special cases in the new framework. The paper sheds new light on fundamental aspects of tensor algebra.