Tensor Train Completion from Fiberwise Observations Along a Single Mode

Tensor completion is an extension of matrix completion aimed at recovering a multiwaydata tensor by leveraging a given subset of its entries (observations) and the patternof observation. The low-rank assumption is key in establishing a relationship betweenthe observed and unobserved entries of the tensor. The low-rank tensor completionproblem is typically solved using numerical optimization techniques, where the rankinformation is used either implicitly (in the rank minimization approach) or explicitly (inthe error minimization approach). Current theories concerning these techniques often studyprobabilistic recovery guarantees under conditions such as random uniform observationsand incoherence requirements. However, if an observation pattern exhibits some lowrankstructure that can be exploited, more efficient algorithms with deterministic recoveryguarantees can be designed by leveraging this structure. This work shows how to useonly standard linear algebra operations to compute the tensor train decomposition of aspecific type of “fiber-wise” observed tensor, where some of the fibers of a tensor (alonga single specific mode) are either fully observed or entirely missing, unlike the usualentry-wise observations. From an application viewpoint, this setting is relevant when it iseasier to sample or collect a multiway data tensor along a specific mode (e.g., temporal).The proposed completion method is fast and is guaranteed to work under reasonabledeterministic conditions on the observation pattern. Through numerical experiments,we showcase interesting applications and use cases that illustrate the effectiveness of theproposed approach.