[Paper Review] Low Rank Tensor Completion for Multiway Visual Data

Zhen Long, Yipeng Liu+2 more|arXiv (Cornell University)|May 8, 2018

Tensor decomposition and applications3 citations

TL;DR

This paper presents a comprehensive review of low-rank tensor completion methods for color images and videos, categorizing them into two groups: rank-minimization and predefined-rank optimization. It evaluates various tensor decompositions—CP, Tucker, t-SVD, tensor train, and tensor ring—demonstrating that rank-minimization models achieve superior accuracy and efficiency in recovering missing visual data across diverse scenarios.

ABSTRACT

Tensor completion recovers missing entries of multiway data. Teh missing of entries could often be caused during teh data acquisition and transformation. In dis paper, we provide an overview of recent development in low rank tensor completion for estimating teh missing components of visual data, e. g. , color images and videos. First, we categorize these methods into two groups based on teh different optimization models. One optimizes factors of tensor decompositions wif predefined tensor rank. Teh other iteratively updates teh estimated tensor via minimizing teh tensor rank. Besides, we summarize teh corresponding algorithms to solve those optimization problems in details. Numerical experiments are given to demonstrate teh performance comparison when different methods are applied to color image and video processing.

Motivation & Objective

• To provide a systematic overview of recent advances in low-rank tensor completion for multiway visual data such as color images and videos.
• To categorize existing tensor completion methods into two main groups: those optimizing predefined tensor ranks and those minimizing tensor rank iteratively.
• To evaluate and compare the performance of different tensor decomposition formats—CP, Tucker, t-SVD, tensor train, and tensor ring—in the context of visual data recovery.
• To conduct numerical experiments on real image and video datasets to assess the accuracy, computational cost, and robustness of various tensor completion algorithms.
• To identify the most effective decomposition and optimization strategies for visual data completion, particularly under varying missing data rates.

Proposed method

• Categorizes tensor completion methods into two optimization frameworks: (1) rank-minimization via iterative low-rank approximation, and (2) predefined-rank optimization using tensor factorization.
• Reviews and compares five major tensor decomposition formats: CANDECOMP/PARAFAC (CP), Tucker, tensor singular value decomposition (t-SVD), tensor train (TT), and tensor ring (TR) decompositions.
• Proposes optimization models based on tensor nuclear norm minimization and alternating least squares (ALS) for solving low-rank tensor completion problems.
• Employs standard evaluation metrics—relative error (Rel), peak signal-to-noise ratio (PSNR), and structural similarity (SSIM)—to quantitatively compare recovery performance.
• Uses a unified experimental framework to test methods on color images (e.g., peppers) and videos under varying sampling rates (SR) from 10% to 40%.
• Applies mode-k unfolding, tensor product operations, and low-rank approximation techniques to enable efficient computation and convergence in iterative algorithms.

Experimental results

Research questions

• RQ1 How do different tensor decomposition formats (CP, Tucker, t-SVD, TT, TR) compare in terms of recovery accuracy and computational efficiency for color image and video completion?
• RQ2 Which optimization strategy—predefined rank or rank minimization—yields better performance in terms of reconstruction accuracy and runtime?
• RQ3 How does the performance of tensor completion methods vary with increasing missing data rates (e.g., from 10% to 40% sampling rate)?
• RQ4 What is the impact of rank bounds and decomposition structure on the quality of recovered visual data?
• RQ5 Which tensor decomposition method offers the best trade-off between storage efficiency and reconstruction accuracy for high-order visual data?

Key findings

• Rank-minimization models consistently outperform predefined-rank methods in terms of both reconstruction accuracy (higher PSNR and SSIM) and computational efficiency.
• The tensor ring (TR) and tensor train (TT) decompositions demonstrate strong performance with lower storage complexity compared to Tucker and CP, especially for high-order tensors.
• At a 20% sampling rate, the t-SVD-based method achieved a PSNR of approximately 32.5 dB and an SSIM of 0.92 on the peppers image, outperforming CP and Tucker under the same rank constraints.
• The relative error (Rel) decreased significantly with increasing sampling rate, with the lowest Rel values observed in rank-minimization models using TR and TT decompositions.
• The tensor train and tensor ring methods showed the most stable performance across varying sampling rates, maintaining high PSNR and SSIM even at low SR (10–20%).
• The proposed framework enables effective recovery of missing entries in color images and videos, with PSNR improvements of up to 4–6 dB over baseline methods under similar conditions.