The geometry of rank-one tensor completion

TL;DR

This paper explores the geometric and algebraic structure of rank-one tensor completion, providing a detailed description of the completable regions, especially for real tensors with specific observation patterns.

Contribution

It offers a comprehensive geometric and semialgebraic analysis of rank-one tensor completion, including boundary descriptions and special cases like diagonal observations.

Findings

Algebraic boundary of the completable region characterized for probability tensors.

Complete semialgebraic description of the completable region for diagonal observations.

Insights into the real tensor rank-one completion problem.

Abstract

The geometry of the set of restrictions of rank-one tensors to some of their coordinates is studied. This gives insight into the problem of rank-one completion of partial tensors. Particular emphasis is put on the semialgebraic nature of the problem, which arises for real tensors with constraints on the parameters. The algebraic boundary of the completable region is described for tensors parametrized by probability distributions and where the number of observed entries equals the number of parameters. If the observations are on the diagonal of a tensor of format $d\times\dots\times d$, the complete semialgebraic description of the completable region is found.