Adaptive Higher-order Spectral Estimators

TL;DR

This paper introduces adaptive higher-order spectral estimators for tensor data, which shrink or threshold singular values to improve signal recovery, especially when the underlying tensor has low multilinear rank.

Contribution

It generalizes matrix spectral shrinkage methods to tensors, developing new estimators with adaptive tuning via Stein's risk estimate for improved multilinear rank estimation.

Findings

Estimators perform well with low multilinear rank tensors.

Method is competitive even when the tensor is not low rank.

Application demonstrated on multivariate relational data.

Abstract

Many applications involve estimation of a signal matrix from a noisy data matrix. In such cases, it has been observed that estimators that shrink or truncate the singular values of the data matrix perform well when the signal matrix has approximately low rank. In this article, we generalize this approach to the estimation of a tensor of parameters from noisy tensor data. We develop new classes of estimators that shrink or threshold the mode-specific singular values from the higher-order singular value decomposition. These classes of estimators are indexed by tuning parameters, which we adaptively choose from the data by minimizing Stein's unbiased risk estimate. In particular, this procedure provides a way to estimate the multilinear rank of the underlying signal tensor. Using simulation studies under a variety of conditions, we show that our estimators perform well when the mean tensor has approximately low multilinear rank, and perform competitively when the signal tensor does not have approximately low multilinear rank. We illustrate the use of these methods in an application to multivariate relational data.