Minimality of tensors of fixed multilinear rank

TL;DR

This paper proves that tensors with fixed multilinear rank form a minimal submanifold in Euclidean space and shows there are no local extrema for linear functionals on rank-one tensors, with applications in statistics.

Contribution

It introduces a geometric property of fixed multilinear rank tensors, demonstrating their minimality and analyzing extremal properties of linear functionals on rank-one tensors.

Findings

Fixed multilinear rank tensors form a minimal submanifold.

No local extrema for linear functionals on rank-one tensors.

Applications in statistical analysis.

Abstract

We discover a geometric property of the space of tensors of fixed multilinear (Tucker) rank. Namely, it is shown that real tensors of fixed multilinear rank form a minimal submanifold of the Euclidean space of tensors endowed with the Frobenius inner product. We also establish the absence of local extrema for linear functionals restricted to the submanifold of rank-one tensors, finding application in statistics.