Tangent Cones to TT Varieties

TL;DR

This paper provides a parametrization and implicit description of the Bouligand tangent cone for tensor varieties of bounded TT rank, generalizing matrix case results and facilitating easier retraction.

Contribution

It introduces a parametrization of the tangent cone for TT tensor varieties and extends the approach to binary hierarchical formats, with explicit and implicit descriptions.

Findings

Parametrization of the Bouligand tangent cone for TT tensor varieties.

Extension of the proof to binary hierarchical formats.

Simplified retraction onto the variety.

Abstract

As already done for the matrix case for example in [Joe Harris, Algebraic Geometry - A first course, p.256] we give a parametrization of the Bouligand tangent cone of the variety of tensors of bounded TT rank. We discuss how the proof generalizes to any binary hierarchical format. The parametrization can be rewritten as an orthogonal sum of TT tensors. Its retraction onto the variety is particularly easy to compose. We also give an implicit description of the tangent cone as the solution of a system of polynomial equations.