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Tensor N-tubal rank and its convex relaxation for low-rank tensor recovery
Yu-Bang Zheng, Ting-Zhu Huang, Xi-Le Zhao, Tai-Xiang Jiang, Teng-Yu Ji, Tian-Hui Ma
TL;DR
Existing t-SVD-based tubal rank is restricted to three-way tensors and is inflexible in representing differing mode correlations. The paper introduces mode-pair unfoldings, N-tubal rank, and its WSTNN relaxation for tensor completion and robust PCA; reported experiments show improved LRTC comparisons and the conclusion emphasizes all-mode correlation modeling.
Problem
T-spectral tubal rank is applicable only to three-way tensors and is inflexible for handling different correlations along different modes.
Source: §1 · p. 3Method
The paper defines mode-k1k2 unfoldings and a vector of their tubal ranks, then uses their weighted TNN sum as the WSTNN relaxation for LRTC and TRPCA.
Source: §1 · pp. 3–4Results
Numerical results indicate that the method exploits correlations across all modes while preserving intrinsic structure, and it performs overall better than compared LRTC methods across all evaluation indices.
Source: §5.1 · p. 16, §6 · p. 22Takeaways & Limitations
The paper concludes that N-tubal rank provides flexible, simultaneous depiction of correlations along different modes compared with tensor tubal rank.
Source: §1 · p. 4
Abstract
from arXiv · showhide
As low-rank modeling has achieved great success in tensor recovery, many research efforts devote to defining the tensor rank. Among them, the recent popular tensor tubal rank, defined based on the tensor singular value decomposition (t-SVD), obtains promising results. However, the framework of the t-SVD and the tensor tubal rank are applicable only to three-way tensors and lack of flexibility to handle different correlations along different modes. To tackle these two issues, we define a new tensor unfolding operator, named mode-$k_1k_2$ tensor unfolding, as the process of lexicographically stacking the mode-$k_1k_2$ slices of an $N$-way tensor into a three-way tensor, which is a three-way extension of the well-known mode-$k$ tensor matricization. Based on it, we define a novel tensor rank, the tensor $N$-tubal rank, as a vector whose elements contain the tubal rank of all mode-$k_1k_2$ unfolding tensors, to depict the correlations along different modes. To efficiently minimize the proposed $N$-tubal rank, we establish its convex relaxation: the weighted sum of tensor nuclear norm (WSTNN). Then, we apply WSTNN to low-rank tensor completion (LRTC) and tensor robust principal component analysis (TRPCA). The corresponding WSTNN-based LRTC and TRPCA models are proposed, and two efficient alternating direction method of multipliers (ADMM)-based algorithms are developed to solve the proposed models. Numerical experiments demonstrate that the proposed models significantly outperform the compared ones.
1. Introduction
The introduction identifies a gap: t-SVD-based tubal rank is limited to three-way tensors and offers little flexibility for modeling different mode correlations. It proposes mode-pair unfoldings, N-tubal rank, and WSTNN, then applies WSTNN to tensor completion and robust PCA.
1. Introduction
- TNN-based t-SVD methods are limited to three-way tensors and do not flexibly model different correlations across modes.Under t-SVD, the first two modes are characterized by SVD while the third is encoded by circular convolution; the paper notes that real-world modes can have unequal correlations.Source: §1 · pp. 2–3
- Mode-k1k2 unfolding lexicographically stacks an N-way tensor's mode-pair slices into a three-way tensor.This extends mode-k tensor matricization to a three-way tensor representation.Source: §1 · p. 3
- Tensor N-tubal rank is a vector containing the tubal ranks of all mode-k1k2 unfolding tensors.It is designed to characterize correlations along different modes more flexibly.Source: §1 · p. 3
- Compared with tensor tubal rank, N-tubal rank flexibly and simultaneously depicts correlations along different modes, as illustrated by rank estimates for two HSIs.The paper reports that the strong third-mode correlation is inadequately represented by the first N-tubal-rank element but captured by the other two.Source: §1 · p. 4, Table 1 · p. 4
- WSTNN relaxes N-tubal rank by taking a weighted sum of the TNNs of the mode-k1k2 unfolding tensors.The weights are nonnegative and sum to one.Source: §1 · p. 4
- The authors propose WSTNN-based LRTC and TRPCA models and develop two ADMM-based algorithms to solve them.Numerical experiments on synthetic and real-world data are used to evaluate the methods.Source: §1 · p. 4
2. Notations and preliminaries
The preliminaries establish tensor notation and mode-k matricization, then introduce the t-product, t-SVD, tubal and multi rank, and tensor nuclear norm.
2. Notations and preliminaries
- Vectors, matrices, and tensors use bold lowercase, uppercase, and calligraphic notation, respectively; the text also specifies tensor norms and vectorization conventions.The Frobenius norm and ℓ1 norm are defined, and an N-way tensor is vectorized into a vector of its entries.Source: §2 · p. 4
- Mode-k matricization maps an N-way tensor to a matrix, with unfold and fold denoting the operation and its inverse.Source: §2 · p. 5
- The t-product replaces scalar multiplication in matrix multiplication with circular convolution between tensor tubes, equivalent to multiplication in the Fourier domain.Source: §2 · p. 5
- For three-way tensors, t-SVD is computed through matrix SVDs in the Fourier domain; tubal rank is the maximum multi-rank entry, and TNN sums Fourier-slice singular values.Multi rank records the rank of each frontal slice after applying the DFT along each tube.Source: §2 · p. 6
3. Tensor N-tubal rank and convex relaxation
The section defines pairwise mode unfolding and uses the tubal ranks of these unfoldings to describe correlations across modes. It then introduces WSTNN as a convex relaxation of this rank.
3. Tensor N-tubal rank and convex relaxation
- Mode-k1k2 unfolding stacks the tensor’s mode-k1k2 slices in lexicographic order into a three-way tensor.The resulting tensor has dimensions n_k1 × n_k2 × the product of the remaining mode sizes.Source: §3 · p. 7
- The N-tubal rank is a vector containing the tubal ranks of all mode-k1k2 unfolding tensors.For a three-way tensor, its tensor tubal rank is the first element of this vector.Source: §3 · p. 8
- The Washington DC Mall example illustrates the low N-tubal-rank prior quantitatively and visually.Source: §3 · p. 8
- Pairwise unfolding avoids losing structure along the second selected mode, while considering all mode pairs captures correlations across all modes.Each individual pairwise tubal rank directly characterizes the two modes in that unfolding, not the other modes.Source: §3 · p. 9
- For a tensor with CP rank r, every N-tubal-rank entry is at most r, and all entries equal r under the stated linear-independence and nonzero Fourier-coefficient conditions.Source: §3 · p. 9
- WSTNN is defined as a weighted sum of the tensor nuclear norms of all mode-k1k2 unfolding tensors, providing a convex relaxation of N-tubal rank.Source: §3 · pp. 9–10
- For HSIs/MSIs, the authors empirically choose weights (θ, 1, 1)/(2 + θ), reflecting the stated stronger spectral-mode correlation.The passage identifies θ as a balance parameter.Source: §3 · p. 10
4. WSTNN-based models and solving algorithms
The paper applies WSTNN to low-rank tensor completion and robust principal component analysis, with ADMM-based algorithms for both models. Each approach uses tensor unfolding and thresholding steps to update its variables.
4.1. WSTNN-based LRTC model
- WSTNN-based LRTC estimates missing tensor entries by minimizing the weighted sum of tensor nuclear norms while enforcing agreement with observed entries.ADMM alternates updates for unfolding-specific auxiliary tensors, the recovered tensor, and Lagrange multipliers.Source: §4.1 · pp. 10–11
- The LRTC auxiliary-tensor updates use tensor singular value thresholding, with the threshold parameter controlling singular-value shrinkage.The algorithm outputs the completed tensor after iterating until convergence or the iteration limit.Source: §4.1 · pp. 11–12
4.2. WSTNN-based TRPCA model
- WSTNN-based TRPCA separates a corrupted observation into a low-rank component and a sparse component, balancing them with a tuning parameter.Its ADMM solver updates the low-rank and sparse components along with auxiliary variables and multipliers.Source: §4.2 · pp. 12–13
- The TRPCA solver uses tensor soft thresholding for an update and outputs the low-rank and sparse components.The algorithm iterates until convergence or the iteration limit.Source: §4.2 · p. 14
5. Numerical experiments
The experiments compare WSTNN-based completion and denoising with competing methods on synthetic tensors and real-world data. Across the reported tests, the authors find stronger recovery results, and examine parameter sensitivity and convergence.
5.1. Low-rank tensor completion
- WSTNN-based LRTC is more robust and preferable than TNN-based LRTC on synthetic three- and four-way tensors across tested N-tubal ranks and sampling rates.Figure 4 compares success rates over varying N-tubal rank and sampling rate.Source: §5.1 · p. 15, Figure 4 · p. 15
- On the 32 CAVE MSIs, WSTNN-based LRTC significantly outperforms competitors across evaluation indices and performs better across all tested cases.The reported visual comparisons also show recovery of shape structure and texture information.Source: §5.1 · p. 16, Table 3 · p. 16, Figure 5 · p. 16
- For Washington DC Mall and Pavia University HSIs, WSTNN-based LRTC achieves the highest values across evaluation indices.The reported Washington DC Mall band comparisons also show the best performance across bands.Source: §5.1 · pp. 16–18, Table 4 · p. 18
- WSTNN-based LRTC significantly outperforms competing methods on MRI and has overall better evaluation-index performance on color-video completion.For HSV completion, it achieves the highest evaluation-index values across sampling rates, with around 4 dB PSNR gain over the second-best method.Source: §5.1 · pp. 18–19, Table 7 · p. 20
5.2. Tensor robust principal component analysis
- WSTNN-based TRPCA is more robust and preferable than TNN-based TRPCA on synthetic three- and four-way tensors across tested ranks and noise levels.On Washington DC Mall and Pavia University HSIs, it performs better than competing methods across evaluation measures and visual comparisons.Source: §5.2 · pp. 20–21, Figure 12 · p. 21, Table 9 · p. 21
5.3. Parameter and convergence analysis
- For both completion and denoising, excessively large or small threshold parameters yield poor results, while moderate values perform best.The authors set τ = (100, 100, 100) for their real-world experiments.Source: §5.3 · p. 22, Figure 14 · p. 22
- The authors state that both ADMM algorithms are theoretically guaranteed to converge because their objectives are convex, and report visual empirical convergence.The empirical convergence plots use τ = (100, 100, 100).Source: §5.3 · p. 22, Figure 14 · p. 22
6. Conclusions
The conclusion presents mode-k1k2 unfolding and the tensor N-tubal rank as a way to characterize correlations across tensor modes, with WSTNN as a convex relaxation. The authors report applying this approach to completion and robust PCA, with numerical results indicating effective use of all-mode correlations while preserving tensor structure.
6. Conclusions
- Mode-k1k2 unfolding reorders an N-way tensor as a three-way tensor, and t-SVD on each such unfolding depicts correlations along different modes.The resulting tensor N-tubal rank has WSTNN as its convex relaxation.Source: §6 · p. 22
- The authors apply WSTNN to low-rank tensor completion and robust PCA, and develop ADMM-based algorithms for both models.They describe the numerical results as demonstrating effective use of correlations along all modes while preserving the underlying tensor's intrinsic structure.Source: §6 · p. 22
- The conclusion attributes the method's supported scope to exploiting correlations along all modes while preserving the underlying tensor's intrinsic structure.This is the authors' stated conclusion from their numerical results.Source: §6 · p. 22