Operator Norm Inequalities between Tensor Unfoldings on the Partition Lattice

TL;DR

This paper investigates how tensor unfolding affects spectral and Frobenius norms, providing inequalities and invariance results that enhance understanding of tensor properties in data analysis.

Contribution

It introduces general inequalities for all tensor unfoldings based on partition lattice, extending beyond traditional matricizations, and identifies conditions for spectral norm invariance.

Findings

Spectral norm of a tensor is bounded by its unfoldings' spectral norms.

Derived an improved upper bound on the Frobenius-to-spectral norm ratio.

Spectral norm remains invariant under certain unfoldings for structured tensors.

Abstract

Interest in higher-order tensors has recently surged in data-intensive fields, with a wide range of applications including image processing, blind source separation, community detection, and feature extraction. A common paradigm in tensor-related algorithms advocates unfolding (or flattening) the tensor into a matrix and applying classical methods developed for matrices. Despite the popularity of such techniques, how the functional properties of a tensor changes upon unfolding is currently not well understood. In contrast to the body of existing work which has focused almost exclusively on matricizations, we here consider all possible unfoldings of an order-$k$ tensor, which are in one-to-one correspondence with the set of partitions of $\{1,\ldots,k\}$. We derive general inequalities between the $l^p$-norms of arbitrary unfoldings defined on the partition lattice. In particular, we demonstrate how the spectral norm ($p=2$) of a tensor is bounded by that of its unfoldings, and obtain an improved upper bound on the ratio of the Frobenius norm to the spectral norm of an arbitrary tensor. For specially-structured tensors satisfying a generalized definition of orthogonal decomposability, we prove that the spectral norm remains invariant under specific subsets of unfolding operations.