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Multi-dimensional imaging data recovery via minimizing the partial sum of tubal nuclear norm
Tai-Xiang Jiang, Ting-Zhu Huang, Xi-Le Zhao, Liang-Jian Deng
TL;DR
Extending low-rank definitions from matrices to tensors remains an open problem, and matrix nuclear-norm minimization can introduce bias. This paper proposes PSTNN as a surrogate for tensor tubal multi-rank and develops algorithms for its recovery models; experiments on varied data illustrate the methods’ effectiveness.
Problem
Extending the low-rank definition from matrices to tensors remains an open problem, while matrix nuclear-norm minimization can cause estimation bias.
Source: §1 · p. 2Method
The paper proposes PSTNN as a surrogate for tensor tubal multi-rank and develops algorithms using a PSVT solver for PSTNN-based recovery models.
Source: §5 · p. 21Results
Experiments on data of various types illustrate the effectiveness of the proposed PSTNN-based methods.
Source: §5 · p. 21Takeaways & Limitations
PSTNN-based methods are presented as an effective approach to tensor recovery across data types.
Source: §5 · p. 21The proposed algorithms’ convergence has not been proved with a theoretical guarantee.
Source: §4.1.3 · p. 16
Abstract
from arXiv · showhide
In this paper, we investigate tensor recovery problems within the tensor singular value decomposition (t-SVD) framework. We propose the partial sum of the tubal nuclear norm (PSTNN) of a tensor. The PSTNN is a surrogate of the tensor tubal multi-rank. We build two PSTNN-based minimization models for two typical tensor recovery problems, i.e., the tensor completion and the tensor principal component analysis. We give two algorithms based on the alternating direction method of multipliers (ADMM) to solve proposed PSTNN-based tensor recovery models. Experimental results on the synthetic data and real-world data reveal the superior of the proposed PSTNN.
1. Introduction
Extending low-rank modeling from matrices to tensors remains an open problem, and nuclear-norm minimization can bias singular values. The paper introduces PSTNN-based recovery models and optimization methods for tensor completion and robust principal component analysis.
1. Introduction
- The motivation is that extending low-rank definitions from matrices to tensors remains an open problem, while matrix nuclear-norm minimization can bias estimates.The cited bias examples are reduced estimated variance and results that may have lower rank than the original data.Source: §1 · p. 2
- PSTNN shrinks small singular values while leaving large ones untouched, yielding low tubal multi-rank estimates without rank-deficiency situations.The paper describes PSTNN as consisting only of small singular values.Source: §1 · p. 2
- The paper proposes PSTNN as a surrogate for tensor tubal multi-rank and extends partial singular value thresholding to exactly solve its minimization problem.The partial singular value thresholding operator was originally proposed for matrices in the complex field.Source: §1 · p. 3
- Two PSTNN-based models address tensor completion and tensor robust principal component analysis, with two ADMM algorithms using the PSVT solver.The paper presents these as typical tensor recovery problems.Source: §1 · p. 3
- Experiments on synthetic and real-world data illustrate that the proposed methods can effectively handle tensor recovery problems.Source: §1 · p. 3
2. Notation and preliminaries
This section establishes notation for tensors and their elements, along with the Fourier transform used for third-order tensors. It also points readers to the t-SVD framework definitions and a table of tensor notations.
2. Notation and preliminaries
- Definitions related to the t-SVD framework are provided in the appendix, while Table 1 lists tensor notations and the tubal nuclear norm is identified as TNN.The section also refers readers to Table 2 for the paper's notations.Source: §2 · pp. 3–4, Table 1 · p. 4
- Lowercase letters, bold lowercase letters, bold uppercase letters, and bold calligraphic letters denote scalars, vectors, matrices, and tensors, respectively.Tensor elements are indexed by their mode indices.Source: §2 · p. 3
- The section defines tensor inner products and the Frobenius norm, and denotes the Fourier transform and its inverse along the third mode.The transform is written as fft(X, [], 3), with the inverse written as ifft.Source: §2 · pp. 3–4
3. Main results
The section defines PSTNN as a surrogate for tubal multi-rank and presents methods for PSTNN-based tensor completion and robust principal component analysis, including an ADMM algorithm for RPCA that produces low-rank and sparse outputs.
3.1. Partial sum of the tubal nuclear norm (PSTNN)
- PSTNN extends partial sums of singular values to the t-SVD framework and serves as a surrogate for the tensor tubal multi-rank.Its definition retains an explicit t-SVD interpretation, and the paper relates it to the tubal multi-rank of the block-diagonal unfolding.Source: §3.1 · p. 5
3.2. The PSTNN-based minimization problem
- The general PSTNN minimization is solved by extending the PSSV-based PSVT scheme to complex matrices and decoupling the tensor problem into Fourier-domain matrix problems.The proposed complex-matrix extension rests on a theorem restated from prior work and extended to the complex case.Source: §3.2 · pp. 6–7
3.3. Tensor completion using PSTNN
- The PSTNN tensor-completion model recovers a tensor while enforcing agreement with observed entries, and ADMM updates use the PSVT-based solver.The observed-entry constraint is represented by the feasible set, while the X-subproblem is solved by Algorithm 1.Source: §3.3 · pp. 8–9
3.4. Tensor RPCA using PSTNN
- PSTNN tensor-RPCA targets recovery of low-rank tensors from sparsely corrupted observations; E denotes sparse corruptions, whose entrywise ℓ1 norm is minimized to promote sparsity, and corresponding subproblems are solved using ADMM.Source: §3.4 · pp. 10–11
- Algorithm 3 solves the PSTNN-based TRPCA model with ADMM, using soft-thresholding to obtain the solution of (23).The algorithm takes the observed tensor, a given tubal multi-rank, and parameter λ as inputs.Source: §3.4 · pp. 11–12
- The procedure initializes L^0 to the observed tensor and E^0 and M^0 to zero tensors, then outputs low-PSTNN L and sparse E.Source: §3.4 · p. 12
4. Experiments
Experiments compare PSTNN with TNN and other recovery methods on synthetic and real-world tensor completion and robust PCA tasks. The reported results favor PSTNN in recovery robustness and reconstruction quality, while the initialization study reports solutions near the ground truth despite the lack of a theoretical convergence guarantee.
4. Experiments
- Experiments were conducted only on third-order tensors; the paper notes that the t-SVD framework extends to arbitrary dimensions.The experiments use PSNR and SSIM, for which larger values correspond to higher reconstruction quality.Source: §4 · pp. 12–13
- The experiments assume N is known for synthetic data and estimate it from the largest 1% singular values of the first Fourier-transformed slice when clean data is available.When clean data is unavailable, the authors recommend rank-decreasing or rank-increasing schemes that adjust an overestimated or underestimated N.Source: §4 · p. 13
4.1. Synthetic data
- Synthetic ground-truth tensors were generated as A = P ∗ Q from independently sampled Gaussian tensors, yielding tubal multi-rank [r, r, · · · , r]ᵀ.The factors have dimensions n1 × r × n3 and r × n2 × n3.Source: §4.1 · p. 13
4.1.1. Tensor completion
- For synthetic tensor completion across sizes, tubal multi-ranks, and sampling rates, PSTNN was more robust than TNN.Each case was repeated 10 times, and a recovery counted as successful when the relative squared error was below 10^-3.Source: §4.1.1 · p. 13
4.1.2. Tensor robust principal components analysis
- In tensor-completion tests varying sparsity and tubal multi-rank, PSTNN-based TC was more robust than TNN-based TC, as indicated by smaller blue areas in Figure 3.Each case was repeated 10 times; each Figure 3 cell reports success percentage, computed by dividing successful times by 10.Source: Figure 3 · p. 15, §4.1.2 · p. 16
4.1.3. Sensitivity to initialization
- Across 1,000 random initializations for a 25 × 25 × 30 tensor with tubal multi-rank 5 and 10% missing entries, solution errors concentrated near the ground truth.The authors note that convergence lacks a theoretical guarantee.Source: §4.1.3 · p. 16
4.2. Tensor completion for the real-world data
- On real-world completion data, PSTNN produced higher visual quality than HaLRTC and TNN and the best PSNR and SSIM results.The experiments used an 80% missing-entry ratio.Source: §4.2 · p. 17, Table 2 · p. 17
4.3. Tensor robust principal components analysis for the color image recovery
- PSTNN-based TRPCA obtained the best image-recovery results, with higher visual quality and better detail and texture preservation than the compared methods.The paper also reports that TNN-based TRPCA achieved the second-best place with respect to PSNR and SSIM.Source: §4.3 · p. 19, Table 3 · p. 19
5. Conclusions
The paper proposes PSTNN as a surrogate for tensor tubal multi-rank and develops PSTNN-based recovery models for tensor completion and tensor robust principal component analysis. ADMM algorithms using a PSVT solver address these models, whose effectiveness is illustrated by experiments on varied data.
5. Conclusions
- PSTNN is proposed within the t-SVD framework as a surrogate for tensor tubal multi-rank.Source: §5 · p. 21
- PSTNN-based minimization models address tensor completion and tensor robust principal component analysis, with ADMM algorithms using a PSVT solver to solve them.The paper describes the PSVT solver as fundamental to solving the PSTNN-based recovery models.Source: §5 · p. 21
- Experiments on data of various types illustrate the effectiveness of the proposed PSTNN-based methods.Source: §5 · p. 21
6. Appendix
The appendix presents definitions used in the t-SVD framework, including tensor products, decompositions, and rank and norm measures. It also proves Theorem 3.1 by reducing an optimization over matrix factors to singular values, yielding the PSVT result.
6.1. The definitions in the t-SVD framework
- The t-product uses circular convolution between tubes, and t-SVD can be computed by frontal-slice SVD after a Fourier transform.A tensor can also be viewed as a matrix whose basic units are tubes.Source: §6.1 · pp. 22–23
- Tubal multi-rank records the ranks of the Fourier-transformed tensor’s frontal slices, while TNN sums singular values across frontal slices.Source: §6.1 · pp. 23–24
6.2. The proof to Theorem 3.1
- The proof parameterizes the optimization using singular values and shows its minimum corresponds to the PSVT result.The singular values are sorted in non-increasing order, and the reduced objective consists of independent quadratic equations for each singular value.Source: §6.2 · pp. 24–25