Note: low-rank tensor train completion with side information based on Riemannian optimization

Abstract

We consider the low-rank tensor train completion problem when additional side information is available in the form of subspaces that contain the mode-k fiber spans. We propose an algorithm based on Riemannian optimization to solve the problem. Numerical experiments show that the proposed algorithm requires far fewer known entries to recover the tensor compared to standard tensor train completion methods.

Variational Bayesian inference for CP tensor completion with side information

“…Indeed, a rank-3 CP tensor of size 300×300×300 can be completed from 1% of its elements, and only 0.004% are needed when 30-dimensional side-information subspaces are available for all of its fibers (columns, rows, and tubes); for a larger 1000 × 1000 × 1000 tensor with the same side information this reduces to 0.0001%. Similar behavior has been observed for Riemannian TT completion with side information [75].…”

“…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.…”

“…Looking closely at Figs. 1 and 2, we can note that the phase transition curves have a bump for small values of n (in fact, it is seen in the phase plots for TT completion as well [75] but remained unnoticed). We explore this phenomenon by zooming in on the phase plots for d = 3, r = 3, and different values of m (Fig.…”

“…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].…”

Tensor train completion: local recovery guarantees via Riemannian optimization

“…This behavior is further well-aligned with the numerical experiments carried out in [40], where a modified RTTC algorithm [33] was introduced to solve (25). Fig.…”