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Variational Bayesian Inference for Tensor Robust Principal Component Analysis

Chao Wang, Huiwen Zheng, Raymond Chan, Youwei Wen

TL;DR

The paper tackles recovering a low-rank tensor from observations corrupted by sparse outliers and Gaussian noise by formulating a Bayesian TRPCA model. It embeds a tensor nuclear norm prior for the low-rank component and a Laplace prior for the sparse part, with Gamma hyperpriors to automatically adjust regularization penalties. Variational Bayesian inference with a mean-field approximation and Laplace-based Gaussian surrogates yields coordinate ascent updates, including tensor SVT for L and soft-thresholding for S, plus extensions to weighted tensor norms and uncertainty quantification. Empirical results on synthetic data, image denoising, and background modeling demonstrate robust recovery, automatic hyperparameter selection, and competitive performance against state-of-the-art TRPCA methods, with credible intervals illustrating reliable uncertainty estimates.

Abstract

Tensor Robust Principal Component Analysis (TRPCA) holds a crucial position in machine learning and computer vision. It aims to recover underlying low-rank structures and to characterize the sparse structures of noise. Current approaches often encounter difficulties in accurately capturing the low-rank properties of tensors and balancing the trade-off between low-rank and sparse components, especially in a mixed-noise scenario. To address these challenges, we introduce a Bayesian framework for TRPCA, which integrates a low-rank tensor nuclear norm prior and a generalized sparsity-inducing prior. By embedding the priors within the Bayesian framework, our method can automatically determine the optimal tensor nuclear norm and achieve a balance between the nuclear norm and sparse components. Furthermore, our method can be efficiently extended to the weighted tensor nuclear norm model. Experiments conducted on synthetic and real-world datasets demonstrate the effectiveness and superiority of our method compared to state-of-the-art approaches.

Variational Bayesian Inference for Tensor Robust Principal Component Analysis

TL;DR

The paper tackles recovering a low-rank tensor from observations corrupted by sparse outliers and Gaussian noise by formulating a Bayesian TRPCA model. It embeds a tensor nuclear norm prior for the low-rank component and a Laplace prior for the sparse part, with Gamma hyperpriors to automatically adjust regularization penalties. Variational Bayesian inference with a mean-field approximation and Laplace-based Gaussian surrogates yields coordinate ascent updates, including tensor SVT for L and soft-thresholding for S, plus extensions to weighted tensor norms and uncertainty quantification. Empirical results on synthetic data, image denoising, and background modeling demonstrate robust recovery, automatic hyperparameter selection, and competitive performance against state-of-the-art TRPCA methods, with credible intervals illustrating reliable uncertainty estimates.

Abstract

Tensor Robust Principal Component Analysis (TRPCA) holds a crucial position in machine learning and computer vision. It aims to recover underlying low-rank structures and to characterize the sparse structures of noise. Current approaches often encounter difficulties in accurately capturing the low-rank properties of tensors and balancing the trade-off between low-rank and sparse components, especially in a mixed-noise scenario. To address these challenges, we introduce a Bayesian framework for TRPCA, which integrates a low-rank tensor nuclear norm prior and a generalized sparsity-inducing prior. By embedding the priors within the Bayesian framework, our method can automatically determine the optimal tensor nuclear norm and achieve a balance between the nuclear norm and sparse components. Furthermore, our method can be efficiently extended to the weighted tensor nuclear norm model. Experiments conducted on synthetic and real-world datasets demonstrate the effectiveness and superiority of our method compared to state-of-the-art approaches.

Paper Structure

This paper contains 30 sections, 83 equations, 5 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations
  4. T-product and t-SVD
  5. Probability distribution
  6. Bayesian model
  7. The likelihood
  8. The prior distributions
  9. Prior distribution for $\mathcal{S}$
  10. Prior distribution for $\mathcal{L}$
  11. The hyper-prior distribution
  12. Joint distribution
  13. Variational Bayesian inference
  14. Kullback-Leibler divergence and evidence lower bound
  15. Mean-field variational family
  16. ...and 15 more sections

Figures (5)

  • Figure 1: Empirical evidence on convergence. Left: objective function, middle: RMSE, right: parameters: $\theta_1, \theta_2$, and $\theta_3$, generated by \ref{['alg:paraalgmain']} across iterations.
  • Figure 2: Uncertainty quantification: recovery of $\overline{\mathcal{L}}_{:ij}$ with 99.73% credible interval (shaded area) where $i = 20, j = 5, 10, 25$.
  • Figure 3: Comparison of color image Gaussian noise removal performance on four examples.
  • Figure 4: Comparison of color image mixed noise removal performance on four examples.
  • Figure 5: Background modeling results of two surveillance video sequences.

Theorems & Definitions (5)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Remark 4.1