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Guaranteed Sampling Flexibility for Low-tubal-rank Tensor Completion

Bowen Su, Juntao You, HanQin Cai, Longxiu Huang

TL;DR

This work introduces tensor Cross-Concentrated Sampling (t-CCS), a flexible sampling model for low-tubal-rank tensor completion that blends ideas from Bernoulli and t-CUR sampling. It establishes a comprehensive recovery theory with explicit constants, showing that a tubal-rank tensor can be uniquely recovered from cross-concentrated samples under reasonable incoherence and sampling requirements. An efficient non-convex solver, Iterative t-CUR Tensor Completion (ITCURTC), is developed to exploit the t-CCS structure with a per-iteration cost of $O(r|I|n_2n_3 + r|J|n_1n_3)$. Extensive synthetic and real-data experiments (color images, MRI, seismic data) demonstrate that ITCURTC achieves reconstruction quality comparable to Bernoulli-based methods while offering substantial runtime advantages, highlighting the practical appeal of t-CCS. The results also reveal avenues for future work, including tighter sampling bounds, convergence analysis of ITCURTC, robustness to noise, and extensions to higher-order tensors.$

Abstract

While Bernoulli sampling is extensively studied in tensor completion, t-CUR sampling approximates low-tubal-rank tensors via lateral and horizontal subtensors. However, both methods lack sufficient flexibility for diverse practical applications. To address this, we introduce Tensor Cross-Concentrated Sampling (t-CCS), a novel and straightforward sampling model that advances the matrix cross-concentrated sampling concept within a tensor framework. t-CCS effectively bridges the gap between Bernoulli and t-CUR sampling, offering additional flexibility that can lead to computational savings in various contexts. A key aspect of our work is the comprehensive theoretical analysis provided. We establish a sufficient condition for the successful recovery of a low-rank tensor from its t-CCS samples. In support of this, we also develop a theoretical framework validating the feasibility of t-CUR via uniform random sampling and conduct a detailed theoretical sampling complexity analysis for tensor completion problems utilizing the general Bernoulli sampling model. Moreover, we introduce an efficient non-convex algorithm, the Iterative t-CUR Tensor Completion (ITCURTC) algorithm, specifically designed to tackle the t-CCS-based tensor completion. We have intensively tested and validated the effectiveness of the t-CCS model and the ITCURTC algorithm across both synthetic and real-world datasets.

Guaranteed Sampling Flexibility for Low-tubal-rank Tensor Completion

TL;DR

This work introduces tensor Cross-Concentrated Sampling (t-CCS), a flexible sampling model for low-tubal-rank tensor completion that blends ideas from Bernoulli and t-CUR sampling. It establishes a comprehensive recovery theory with explicit constants, showing that a tubal-rank tensor can be uniquely recovered from cross-concentrated samples under reasonable incoherence and sampling requirements. An efficient non-convex solver, Iterative t-CUR Tensor Completion (ITCURTC), is developed to exploit the t-CCS structure with a per-iteration cost of . Extensive synthetic and real-data experiments (color images, MRI, seismic data) demonstrate that ITCURTC achieves reconstruction quality comparable to Bernoulli-based methods while offering substantial runtime advantages, highlighting the practical appeal of t-CCS. The results also reveal avenues for future work, including tighter sampling bounds, convergence analysis of ITCURTC, robustness to noise, and extensions to higher-order tensors.$

Abstract

While Bernoulli sampling is extensively studied in tensor completion, t-CUR sampling approximates low-tubal-rank tensors via lateral and horizontal subtensors. However, both methods lack sufficient flexibility for diverse practical applications. To address this, we introduce Tensor Cross-Concentrated Sampling (t-CCS), a novel and straightforward sampling model that advances the matrix cross-concentrated sampling concept within a tensor framework. t-CCS effectively bridges the gap between Bernoulli and t-CUR sampling, offering additional flexibility that can lead to computational savings in various contexts. A key aspect of our work is the comprehensive theoretical analysis provided. We establish a sufficient condition for the successful recovery of a low-rank tensor from its t-CCS samples. In support of this, we also develop a theoretical framework validating the feasibility of t-CUR via uniform random sampling and conduct a detailed theoretical sampling complexity analysis for tensor completion problems utilizing the general Bernoulli sampling model. Moreover, we introduce an efficient non-convex algorithm, the Iterative t-CUR Tensor Completion (ITCURTC) algorithm, specifically designed to tackle the t-CCS-based tensor completion. We have intensively tested and validated the effectiveness of the t-CCS model and the ITCURTC algorithm across both synthetic and real-world datasets.
Paper Structure (44 sections, 22 theorems, 156 equations, 16 figures, 4 tables, 3 algorithms)

This paper contains 44 sections, 22 theorems, 156 equations, 16 figures, 4 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Main contributions
  3. Related work
  4. Tensor completion in tubal-rank setting
  5. t-CUR decompositions
  6. Notation and preliminaries
  7. Proposed sampling model
  8. Theoretical results
  9. An efficient solver for t-CCS
  10. Iterative t-CUR tensor completion algorithm
  11. Computational complexity analysis
  12. Numerical experiments
  13. Synthetic data examples
  14. Real-world applications
  15. Color image completion
  16. ...and 29 more sections

Key Result

Theorem 1

Let $\mathcal{T} \in \mathbb{K}^{n_1 \times n_2 \times n_3}$ with multi-rank $\textnormal{rank}_m(\mathcal{T})=\vec{r}$. Let $I \subseteq [n_1]$ and $J \subseteq [n_2]$ be two index sets. Denote $\mathcal{C}=[\mathcal{T}]_{:, J,:}$, $\mathcal{R}=[\mathcal{T}]_{I,:,:}$, and $\mathcal{U}=[\mathcal{T}]

Figures (16)

  • Figure 2: Visual results of color image inpainting using t-CCS samples at an overall sampling rate of $20\%$ with BCPF, TMac, TNN, and F-TNN algorithms.
  • Figure 3: (Row 1) 3D and (Row 2) 2D views illustrate ITCURTC's empirical phase transition for the t-CCS model. $\delta=|I|/768=|J|/768$ shows sampled indices ratios, $p$ is the Bernoulli sampling probability over subtensors, and $\alpha$ is the overall tensor sampling rate. White and black in the $768\times 768\times 256$ tensor results represent success and failure, respectively, across 25 tests for tubal ranks 2, 5, and 7 (Columns 1-3). The $\alpha$ needed for success remains consistent across different combinations $\delta$ and $p$.
  • Figure 4: The visualization of color image inpainting for Building and Window datasets by setting tubal-rank $r= 35$ with the percentage of selected horizontal and lateral slices $\delta = 13\%$ and overall sampling rate $20\%$ for ITCURTC. Other algorithms are applied based on Bernoulli sampling model with the same overall sampling rate $20\%$. Additionally, t-CCS samples on the Building for ITCURTC are the same as those in \ref{['failure_pics']}.
  • Figure 5: The visualizations of MRI data recovery are obtained by setting a tubal rank of $r = 35$ for ITCURTC with the percentage of selected lateral and horizontal slices $\delta = 27\%$ at an overall sampling rate of $30\%$. Other algorithms are applied under Bernoulli sampling models with the same overall sampling rate. Results for slices 51, 66, 86, and 106 are shown in rows 1 to 4. Each set includes a $1.3\times$ magnified area for clearer comparison, positioned at the bottom left of each result.
  • Figure 6: Visualization of seismic data recovery results by setting tubal-rank $r = 3$ for ITCURTC with a percentage of the selected horizontal and lateral slices $\delta = 17\%$ and an overall sampling rate $\alpha=28\%$, while other methods are applied based on Bernoulli sampling models with the same overall sampling rate $28\%$. Displayed are slices $15$, $25$, and $35$ from top to bottom, with a $1.2\times$ magnified area in each set for a clearer comparison.
  • ...and 11 more figures

Theorems & Definitions (42)

  • Definition 1: Tubal-rank and multi-rank
  • Theorem 1: weiyimin
  • Definition 2: $f$-diagonal tensor
  • Definition 3: Tensor transpose
  • Definition 4: Identity tensor
  • Definition 5: Orthogonal tensor
  • Definition 6: t-SVD
  • Definition 7: Moore-Penrose inverse
  • Definition 8: Tensor Frobenius norm
  • Definition 9: Standard tensor lateral basis
  • ...and 32 more