# Tensor Train Completion from Fiberwise Observations Along a Single Mode

arXiv: https://arxiv.org/abs/2509.18149 · HTML: https://mlsanity.com/paper/2509.18149/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 58 references that are in the map.
- First posted on arXiv 16 Sep 2025.

## Builds on (most cited first, 11 of 11)

- [Tensor Decomposition for Signal Processing and Machine Learning](https://mlsanity.com/paper/1607.01668/) (2016)
- [Efficient tensor completion for color image and video recovery: Low-rank tensor train](https://mlsanity.com/paper/1606.01500/) (2016)
- [Tensor Completion Algorithms in Big Data Analytics](https://mlsanity.com/paper/1711.10105/) (2017)
- [A Nonconvex Low-Rank Tensor Completion Model for Spatiotemporal Traffic Data Imputation](https://mlsanity.com/paper/2003.10271/) (2020)
- [A Characterization of Deterministic Sampling Patterns for Low-Rank Matrix Completion](https://mlsanity.com/paper/1503.02596/) (2015)
- [High-dimension Tensor Completion via Gradient-based Optimization Under Tensor-train Format](https://mlsanity.com/paper/1804.01983/) (2018)
- [Generalized Canonical Correlation Analysis: A Subspace Intersection Approach](https://mlsanity.com/paper/2003.11205/) (2020)
- [Tensor Train Quantum State Tomography using Compressed Sensing](https://mlsanity.com/paper/2506.23560/) (2025)
- [A case study of spatiotemporal forecasting techniques for weather forecasting](https://mlsanity.com/paper/2209.14782/) (2022)
- [Deterministic Conditions for Subspace Identifiability from Incomplete Sampling](https://mlsanity.com/paper/1410.0633/) (2014)
- [Minimal Rank Completions for Overlapping Blocks](https://mlsanity.com/paper/2106.11267/) (2021)

## Built on by (most cited first, 1 of 1)

- [Tomography by Design: An Algebraic Approach to Low-Rank Quantum States](https://mlsanity.com/paper/2602.15202/) (2026)

## Abstract (arXiv)

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.