Symmetric Embeddings of Tensors, Block Tensors and Block Unfoldings.

Stefan Ragnarsson (CAM)

Block matrices and block methods are ubiquitous is matrix analysis. They allow concise and intuitive derivation of important algorithms and are key to achieving high computational performance. As the field of tensor computations matures, block-based methods will become increasingly important. However, for higher-order tensors the generalizations of block matrix facts like forming the transpose become nontrivial to prove. In this talk a precise definition and notation for a block tensor is given, which allow for relatively simple proofs of basic properties.

With the notion of a block tensor firmly established, the important matrix concept of symmetrically embedding a matrix A into a larger matrix [0 A; A' 0] can be readily generalized to higher-order tensors. It will be shown that this allows us to connect the concepts of tensor eigenvalues and tensor singular values and even generalize algorithms originally derived only for symmetric tensors to general non-symmetric ones.

In the final part of the talk the concept of a block unfolding of a tensor is discussed. When doing tensor calculations most tensor operations are translated into operations on unfoldings (also known as flattenings or matricizations), where the tensor is "unfolded" into a matrix. Unfortunately, standard unfolding schemes do not work well with block tensors, as block elements are scattered through the unfolded matrix. Block unfoldings solve this problem, and it will be shown how all the important tensor operations can be performed by using this scheme. Such results will be key to efficient block tensor computations.