Outlier-aware Tensor Robust Principal Component Analysis with Self-guided Data Augmentation

TL;DR

This work tackles tensor robust PCA (TRPCA) under structured, non-sparse outliers by introducing a self-guided data augmentation framework. It learns a weight tensor $\mathcal{W}$ and an augmented tensor $\mathcal{Y}$ to reformulate TRPCA as standard TPCA, enabling efficient closed-form updates via proximal block coordinate descent and a convergence guarantee to stationary points. A generalized weight scheme $\mathcal{W}_{ijk}=\exp(-\frac{(\mathcal{Y}_{ijk}-\mathcal{X}_{ijk})^2}{2\gamma})$ extends outlier detection beyond binary masks, with $\gamma \to 0$ recovering oracle behavior. Empirical results on synthetic data and real tasks (face denoising, background subtraction, hyperspectral denoising) show improved accuracy and substantial speedups over state-of-the-art methods, validating the framework's practical impact on high-dimensional tensor processing.

Abstract

Tensor Robust Principal Component Analysis (TRPCA) is a fundamental technique for decomposing multi-dimensional data into a low-rank tensor and an outlier tensor, yet existing methods relying on sparse outlier assumptions often fail under structured corruptions. In this paper, we propose a self-guided data augmentation approach that employs adaptive weighting to suppress outlier influence, reformulating the original TRPCA problem into a standard Tensor Principal Component Analysis (TPCA) problem. The proposed model involves an optimization-driven weighting scheme that dynamically identifies and downweights outlier contributions during tensor augmentation. We develop an efficient proximal block coordinate descent algorithm with closed-form updates to solve the resulting optimization problem, ensuring computational efficiency. Theoretical convergence is guaranteed through a framework combining block coordinate descent with majorization-minimization principles. Numerical experiments on synthetic and real-world datasets, including face recovery, background subtraction, and hyperspectral denoising, demonstrate that our method effectively handles various corruption patterns. The results show the improvements in both accuracy and computational efficiency compared to state-of-the-art methods.

Figures (6)

• Figure 1: Correct recovery for different levels of Tucker rank and sparsity. Fraction of correct recoveries across 10 trials, as a function of Tucker rank ($x$-axis) and sparsity of $\mathcal{S}_0$ ($y$-axis).
• Figure 2: Relative error of $\mathcal{L}_t$ versus iteration number for different values of $\lambda$. The left and right plots correspond to $\rho=0.3$ and $\rho=0.6$, respectively.
• Figure 3: Noise and shadows removal from face images of subject 1 (row I), subject 2 (row II), and subject 6 (row III). The values below each method indicate the average running time in seconds (s).
• Figure 4: Visual comparison of background and foreground extraction from four videos. The first column shows the original data and ground truth, while the subsequent columns display the background and foreground extracted by different methods. Each two rows from top to bottom correspond to videos: "blizzard", "office", "skating", and "snowFall", respectively.
• Figure 5: Comparison of weight tensors in background subtraction: (a) Ground truth binary mask (white for outlier, black for clean). (b) Oracle weight (black for outlier, white for clean). (c) Outlier-aware weight estimated by our model.

Theorems & Definitions (9)

• Definition 1: Mode-$k$ matricization cai2024robust
• Definition 2: Mode-$k$ product cai2024robust
• Definition 3: Tucker decomposition and Tucker rank kolda2009tensor
• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• Theorem 1