Tangent Cones to TT Varieties
This work addresses a theoretical problem in algebraic geometry for tensor decompositions, but it appears incremental as it extends known matrix results to tensor formats.
The paper provides a parametrization of the Bouligand tangent cone for tensors with bounded TT rank, generalizing the proof to binary hierarchical formats and offering an implicit description via polynomial equations.
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.