Tensor train completion: local recovery guarantees via Riemannian optimization

Abstract

In this work we estimate the number of randomly selected elements of a tensor that with high probability guarantees local convergence of Riemannian gradient descent for tensor train completion. We derive a new bound for the orthogonal projections onto the tangent spaces based on the harmonic mean of the unfoldings' singular values and introduce a notion of core coherence for tensor trains. We also extend the results to tensor train completion with side information and obtain the corresponding local convergence guarantees.

Variational Bayesian inference for CP tensor completion with side information

“…We would also like to add a few words about the bump in the phase transition curves that we observed for m n. In [75], such bump can be recognized on the phase plot corresponding to Riemannian tensor train completion with side information of a 10-dimensional tensor. At the same time, the results of [76] tell us that Riemannian gradient descent converges locally if the number of samples |Ω| exceeds a certain threshold that depends on m and is independent of n, and this behavior is indeed seen on the phase plots for larger values of n. However, random initialization that is used in [75] (and in this paper too) certainly does not put the initial condition into the basin of local attraction. Recent results on non-convex optimization [82][83][84] show that gradient descent converges globally in certain problems (including matrix completion) when initialized randomly, provided, of course, that |Ω| is large enough.…”

“…Completion and factorization problems with different kinds of auxiliary information have been studied in the literature for CP [66][67][68][69][70][71][72][73] and Tucker [74] decompositions. Side information, as we defined it, received less attention: it was used for TT completion [75,76], Tucker completion [77], and we have not seen such papers for the CP decomposition. Similar formulations appear in kernelized matrix completion [78,79].…”

Rank Estimation for Third-Order Tensor Completion in the Tensor-Train Format

“…Intuitively the smaller |Ω|, the more difficult it is to recover A from A Ω by solving (1). However, the minimal number of samples needed is not known [6].…”