Tensor Train Completion from Fiberwise Observations Along a Single Mode

2025-09-16

AI Summary

This work addresses the tensor completion problem under fiber-wise structured sampling: recovering a tensor exactly when only entire fibers along a single mode are observed, while all other fibers are completely missing. We propose a deterministic completion method based on Tensor Train (TT) decomposition, relying solely on standard linear algebra operations. To our knowledge, this is the first approach to provide rigorous, provable exact reconstruction guarantees for fiber-sampled tensors—breaking away from conventional assumptions of random, entry-wise sampling. The method fully exploits both the low-rank TT prior and the inherent structure of the sampling pattern, achieving both computational efficiency and theoretical soundness. Numerical experiments demonstrate high accuracy and strong robustness on real-world spatiotemporal data, significantly outperforming state-of-the-art tensor completion algorithms.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Reasoning under Uncertainty: Relational Probabilistic Models
• Computer Vision: Other Foundations of Computer Vision

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Bridging structured and unstructured data
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

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.

Problem

• Recovering multiway tensor data from fiberwise observations along single mode
• Developing efficient tensor completion method using linear algebra operations
• Addressing tensor recovery when sampling occurs along specific mode

Innovation

• Fiberwise observations along single mode
• Tensor train decomposition using linear algebra
• Deterministic recovery guarantees for structured patterns