Tensor completion is an extension of matrix completion aimed at recovering a multiway data tensor by leveraging a given subset of its entries (observations) and the pattern of observation. The low-rank assumption is key in establishing a relationship between the observed and unobserved entries of the tensor. The low-rank tensor completion problem is typically solved using numerical optimization techniques, where the rank information is used either implicitly (in the rank minimization approach) or explicitly (in the error minimization approach). Current theories concerning these techniques often study probabilistic recovery guarantees under conditions such as random uniform observations and incoherence requirements. However, if an observation pattern exhibits some low-rank structure that can be exploited, more efficient algorithms with deterministic recovery guarantees can be designed by leveraging this structure. This work shows how to use only standard linear algebra operations to compute the tensor train decomposition of a specific type of ``fiber-wise'' observed tensor, where some of the fibers of a tensor (along a single specific mode) are either fully observed or entirely missing, unlike the usual entry-wise observations. From an application viewpoint, this setting is relevant when it is easier 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 reasonable deterministic conditions on the observation pattern. Through numerical experiments, we showcase interesting applications and use cases that illustrate the effectiveness of the proposed approach.

翻译：张量补全是矩阵补全的扩展，旨在通过利用给定子集的张量条目（观测值）及其观测模式来恢复多维数据张量。低秩假设是建立张量已观测与未观测条目间关系的关键。低秩张量补全问题通常通过数值优化技术求解，其中秩信息或以隐式方式（秩最小化方法）或以显式方式（误差最小化方法）被利用。现有关于这些技术的理论通常研究在随机均匀观测和非相干性要求等条件下的概率恢复保证。然而，若观测模式本身具有可被利用的低秩结构，则可通过利用该结构设计出具有确定性恢复保证的更高效算法。本文展示了如何仅使用标准线性代数运算来计算一类特殊“纤维式”观测张量的张量链分解，其中张量的部分纤维（沿单一特定模式）要么被完全观测，要么完全缺失，这与常见的逐条目观测方式不同。从应用视角看，当沿特定模式（例如时间维度）采样或收集多维数据张量更为便捷时，此设定具有实际意义。所提出的补全方法计算快速，且能在观测模式满足合理确定性条件时保证有效。通过数值实验，我们展示了说明该方法有效性的有趣应用场景和用例。