Tangent cones to tensor train varieties

Abstract

As already done for the matrix case in [1, p. 256], [2, Thm. 6.1, p. 1872] and [3, Thm. 3.2], we give a parametrization of the Bouligand tangent cone of the variety of tensors of bounded tensor train (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.