Improving Approximation Guarantees for Tree Tensor Decompositions

Develop a polynomial-time algorithm for a fixed m-node tree tensor network (e.g., tensor train or Tucker) with prescribed ranks that, for any input tensor A ∈ R^{d_1×⋯×d_n}, computes X in the representable set S such that ||X−A||_F^2 < (m−1)·min_{Y∈S} ||Y−A||_F^2; or alternatively, show under standard complexity-theoretic assumptions that an (m−1)-approximation factor is optimal.

Background

Sequential SVD-based algorithms for tree tensor formats (TT-SVD, HOSVD) guarantee an (m−1)-factor relative to the best representable approximation; this bound is tight in some cases for Tucker.

It remains open whether improved worst-case guarantees are possible in polynomial time for general tree tensor networks, including tensor trains.