Perturbation bounds for (nearly) orthogonally decomposable tensors with statistical applications

Abstract

We develop deterministic perturbation bounds for singular values and vectors of orthogonally decomposable tensors, in a spirit similar to classical results for matrices such as those due to Weyl, Davis, Kahan and Wedin. Our bounds demonstrate intriguing differences between matrices and higher order tensors. Most notably, they indicate that for higher order tensors perturbation affects each essential singular value/vector in isolation, and its effect on an essential singular vector does not depend on the multiplicity of its corresponding singular value or its distance from other singular values. Our results can be readily applied and provide a unified treatment to many different problems involving higher order orthogonally decomposable tensors. In particular, we illustrate the implications of our bounds through connected yet seemingly different high-dimensional data analysis tasks: the unsupervised learning scenario of tensor SVD and the supervised task of tensor regression, leading to new insights in both of these settings.