Sufficient and Necessary Conditions for Eckart-Young like Result for Tubal Tensors
Provides a complete theoretical characterization for tensor decompositions, benefiting researchers in multilinear algebra and tensor-based machine learning.
The paper characterizes the family of tubal products that yield an Eckart-Young type theorem for tensors, showing that the best low-rank approximation under Frobenius norm is obtained by truncating tubal SVD. Experiments on video data and dynamical systems validate the theoretical results.
A valuable feature of the tubal tensor framework is that many familiar constructions from matrix algebra carry over to tensors, including SVD and notions of rank. Importantly, it has been shown that for a specific family of tubal products, an Eckart-Young type theorem holds, i.e., the best low-rank approximation of a tensor under the Frobenius norm is obtained by truncating its tubal SVD. In this paper, we provide a complete characterization of the family of tubal products that yield an Eckart-Young type result. We demonstrate the practical implications of our theoretical findings by conducting experiments with video data and data-driven dynamical systems.