# Smooth PARAFAC Decomposition for Tensor Completion

arXiv: https://arxiv.org/abs/1505.06611 · HTML: https://mlsanity.com/paper/1505.06611/

## Key facts

- Cited by 34 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 25 May 2015.

## Built on by (most cited first, 30 of 34)

- [Implicit Regularization in Tensor Factorization](https://mlsanity.com/paper/2102.09972/) (2021)
- [Guaranteed Tensor Recovery Fused Low-rankness and Smoothness](https://mlsanity.com/paper/2302.02155/) (2023)
- [Low-Rank Autoregressive Tensor Completion for Spatiotemporal Traffic Data Imputation](https://mlsanity.com/paper/2104.14936/) (2021)
- [Permutation Search of Tensor Network Structures via Local Sampling](https://mlsanity.com/paper/2206.06597/) (2022)
- [Alternating Local Enumeration (TnALE): Solving Tensor Network Structure Search with Fewer Evaluations](https://mlsanity.com/paper/2304.12875/) (2023)
- [Effective Tensor Completion via Element-wise Weighted Low-rank Tensor Train with Overlapping Ket Augmentation](https://mlsanity.com/paper/2109.05736/) (2021)
- [Robust Factorization of Real-world Tensor Streams with Patterns, Missing Values, and Outliers](https://mlsanity.com/paper/2102.08466/) (2021)
- [tnGPS: Discovering Unknown Tensor Network Structure Search Algorithms via Large Language Models (LLMs)](https://mlsanity.com/paper/2402.02456/) (2024)
- [Tensor Decomposition Meets RKHS: Efficient Algorithms for Smooth and Misaligned Data](https://mlsanity.com/paper/2408.05677/) (2024)
- [New Riemannian preconditioned algorithms for tensor completion via polyadic decomposition](https://mlsanity.com/paper/2101.11108/) (2021)
- [A case study of spatiotemporal forecasting techniques for weather forecasting](https://mlsanity.com/paper/2209.14782/) (2022)
- [Low-Rank Tensor Completion via Novel Sparsity-Inducing Regularizers](https://mlsanity.com/paper/2310.06233/) (2023)
- [Bayesian Complementary Kernelized Learning for Multidimensional Spatiotemporal Data](https://mlsanity.com/paper/2208.09978/) (2022)
- [Variational Bayesian inference for CP tensor completion with side information](https://mlsanity.com/paper/2206.12486/) (2022)
- [A generalizable framework for low-rank tensor completion with numerical priors](https://mlsanity.com/paper/2302.05881/) (2023)
- [Nonnegative Low-Rank Tensor Completion via Dual Formulation with Applications to Image and Video Completion](https://mlsanity.com/paper/2305.07976/) (2023)
- [Image Reconstruction using Superpixel Clustering and Tensor Completion](https://mlsanity.com/paper/2305.09564/) (2023)
- [Haar Nuclear Norms with Applications to Remote Sensing Imagery Restoration](https://mlsanity.com/paper/2407.08509/) (2024)
- [Generalized Least Squares Kernelized Tensor Factorization](https://mlsanity.com/paper/2412.07041/) (2024)
- [Low Tensor-Rank Adaptation of Kolmogorov--Arnold Networks](https://mlsanity.com/paper/2502.06153/) (2025)
- [Robust Low-Rank Tensor Completion based on M-product with Weighted Correlated Total Variation and Sparse Regularization](https://mlsanity.com/paper/2604.13525/) (2026)
- ["Sparse + Low-Rank'' Tensor Completion Approach for Recovering Images and Videos](https://mlsanity.com/paper/2110.09298/) (2021)
- [Soft Smoothness for Audio Inpainting Using a Latent Matrix Model in Delay-embedded Space](https://mlsanity.com/paper/2203.09746/) (2022)
- [Latent Matrices for Tensor Network Decomposition and to Tensor Completion](https://mlsanity.com/paper/2210.03392/) (2022)
- [Optical Coherence Tomography Image Enhancement via Block Hankelization and Low Rank Tensor Network Approximation](https://mlsanity.com/paper/2306.11750/) (2023)
- [Neural Gradient Regularizer](https://mlsanity.com/paper/2308.16612/) (2023)
- [ADMM-MM Algorithm for General Tensor Decomposition](https://mlsanity.com/paper/2312.11763/) (2023)
- [Functional-Edged Network Modeling](https://mlsanity.com/paper/2404.00218/) (2024)
- [BAMITA: Bayesian Multiple Imputation for Tensor Arrays](https://mlsanity.com/paper/2410.23412/) (2024)
- [MNT-TNN: Spatiotemporal Traffic Data Imputation via Compact Multimode Nonlinear Transform-based Tensor Nuclear Norm](https://mlsanity.com/paper/2503.22955/) (2025)

## Abstract (arXiv)

In recent years, low-rank based tensor completion, which is a higher-order extension of matrix completion, has received considerable attention. However, the low-rank assumption is not sufficient for the recovery of visual data, such as color and 3D images, where the ratio of missing data is extremely high. In this paper, we consider "smoothness" constraints as well as low-rank approximations, and propose an efficient algorithm for performing tensor completion that is particularly powerful regarding visual data. The proposed method admits significant advantages, owing to the integration of smooth PARAFAC decomposition for incomplete tensors and the efficient selection of models in order to minimize the tensor rank. Thus, our proposed method is termed as "smooth PARAFAC tensor completion (SPC)." In order to impose the smoothness constraints, we employ two strategies, total variation (SPC-TV) and quadratic variation (SPC-QV), and invoke the corresponding algorithms for model learning. Extensive experimental evaluations on both synthetic and real-world visual data illustrate the significant improvements of our method, in terms of both prediction performance and efficiency, compared with many state-of-the-art tensor completion methods.