Robust Tensor CUR Decompositions: Rapid Low-Tucker-Rank Tensor Recovery with Sparse Corruption

TL;DR

This work tackles robust tensor recovery under Tucker-rank constraints by formulating TRPCA as decomposing a corrupted tensor $\mathcal{X}$ into a low-Tucker-rank part $\mathcal{L}$ and a sparse part $\mathcal{S}$, solved via a fast non-convex algorithm RTCUR. RTCUR uses alternating projections between sparsity enforcement (hard-thresholding) and inexact low-Tucker-rank approximation implemented through mode-wise tensor CUR decompositions (Fiber and Chidori), yielding significant computational savings over HOSVD-based TRPCA methods. The authors provide theoretical support for the tensor sparsity model, present four sampling variants, and demonstrate through extensive synthetic and real-data experiments that RTCUR is faster and often more robust than state-of-the-art TRPCA methods in tasks such as robust face modeling, video background subtraction, and network clustering. This approach enables scalable, reliable tensor recovery in large-scale applications by exploiting tensor CUR structure and adaptive sparsity handling.

Abstract

We study the tensor robust principal component analysis (TRPCA) problem, a tensorial extension of matrix robust principal component analysis (RPCA), that aims to split the given tensor into an underlying low-rank component and a sparse outlier component. This work proposes a fast algorithm, called Robust Tensor CUR Decompositions (RTCUR), for large-scale non-convex TRPCA problems under the Tucker rank setting. RTCUR is developed within a framework of alternating projections that projects between the set of low-rank tensors and the set of sparse tensors. We utilize the recently developed tensor CUR decomposition to substantially reduce the computational complexity in each projection. In addition, we develop four variants of RTCUR for different application settings. We demonstrate the effectiveness and computational advantages of RTCUR against state-of-the-art methods on both synthetic and real-world datasets.

Figures (8)

• Figure 1: (cai2021mode). Illustration of the Fiber CUR Decomposition of \ref{['thm: CUR Char']} in which $J_i$ is not necessarily related to $I_i$. The lines correspond to rows of $\bm{C}_2$, and red indices correspond to rows of $\bm{U}_2$. Note that the lines may (but do not have to) pass through the core subtensor $\mathcal{R}$ outlined by dotted lines. For the figure's clarity, we do not show fibers in $\bm{C}_1$ and $\bm{C}_3$.
• Figure 1: T-Sparsity vs. M-Sparsity. A black box represents an outlier entry and a white box represents a good entry. The right-hand-side matrix is unfolded from the left-hand-side tensor.
• Figure 1: Empirical phase transition in corruption rate $\alpha$ and sampling constant $\upsilon$. Left to Right: RTCUR-FF, RTCUR-RF, RTCUR-FC, RTCUR-RC, Top: $r = 3$. Middle: $r = 5$. Bottom: $r = 10$.
• Figure 2: Runtime vs. dimension comparison among variants of RTCUR, RGD, IRCUR and AAP on tensors with size $d\times d\times d$ and Tucker rank $(3,3,3)$. The RGD method proceeds relatively slowly for larger tensors, so we only test the RGD runtime for tensors with a size smaller than 300 for each mode.
• Figure 3: Runtime vs. relative error comparison among RTCUR-F, RTCUR-R, AAP, and IRCUR on tensor with size $500\times 500\times 500$ and Tucker rank ($3,3,3$).

Theorems & Definitions (12)

• Definition 1.1: Tensor matricization/unfolding
• Definition 1.2: Mode-$k$ product
• Definition 1.3: Tensor Tucker rank and Tucker decomposition
• Theorem 1.4: cai2021mode
• Remark 1.5
• Definition 2.1: $\alpha$-M-sparsity for matrix
• Definition 2.2: $\alpha$-T-sparsity for tensor
• Theorem 2.3
• Remark 2.4
• Proof 1: Proof of \ref{['asparse']}