Fiber Sampling Approach to Canonical Polyadic Decomposition and Application to Tensor Completion

Abstract

Tensor decompositions play an important role in a variety of applications, such as signal processing and machine learning. In practice, the tensor can be incomplete or very large, making it difficult to analyze and process using conventional tensor techniques. In this paper we focus on the basic canonical polyadic decomposition (CPD). We propose an algebraic framework for finding the CPD of tensors that have missing fibers. This includes extensions of multilinear algebraic as well as generic uniqueness conditions originally developed for the CPD of fully observed tensors. Computationally, we reduce the CPD of a tensor with missing fibers to relatively simple matrix completion problems via a matrix eigenvalue decomposition (EVD). Under the given conditions, the EVD-based algorithm is guaranteed to return the exact CPD. The derivation establishes connections with so-called coupled CPDs, an emerging concept that has proven to be of great interest in a range of array processing and signal processing applications. It will become clear that relatively few fibers are needed in order to compute the CPD. This makes fiber sampling interesting for large scale tensor decompositions. Numerical experiments show that the algebraic framework may significantly speed up more common optimization-based computation schemes for the estimation of the CPD of incomplete noisy data tensors.