Tensor Decomposition Analysis

• Abstract

  A tensor is a multiway array of scalars; in this context it is a generalization of scalars (order-0), vectors (order-1), and matrices (order-2) to higher-order structures. Tensor analysis is an increasingly relevant and interesting field of inquiry in signal processing and machine learning as generalizations of rank-1 decompositions of matrices such as the singular value decomposition (or eigendecomposition of symmetric matrices) have found considerable application and success. Decompositions of tensors emerge and become relevant when high-order data or statistics are analyzed.

• Description

  A tensor is a multiway array of scalars; in this context it is a generalization of scalars (order-0), vectors (order-1), and matrices (order-2) to higher-order structures. Tensor analysis is an increasingly relevant and interesting field of inquiry in signal processing and machine learning as generalizations of rank-1 decompositions of matrices such as the singular value decomposition (or eigendecomposition of symmetric matrices) have found considerable application and success. Decompositions of tensors emerge and become relevant when high-order data or statistics are analyzed. Canonical decomposition (also called parallel factor analysis) and other decomposition methodologies exist and exhibit useful properties such as uniqueness. However, this particular decomposition lacks structure in its vector frame that forms the rank-one decomposition components, which prevents recursive solution formulations as in deflation of principal components. The goal of this project is to develop a non-redundant tensor decomposition, which can be interpreted as a generalization of the spherical coordinate system of vectors and the orthogonal matrix group based on a predetermined frame of basis vectors. This imposed structure prevents the decomposition from achieving the minimum cross-product tensor rank as canonical decomposition does; however, we think it provides a solution that is more suitable for recursive calculations that is useful and important in dimension reduction applications.

• Team Members