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Robust Tensor Principal Component Analysis: Exact Recovery via Deterministic Model

Bo Shen, Yutong Zhang, Zhenyu, Kong

TL;DR

This work establishes deterministic exact recovery guarantees for Robust Tensor Principal Component Analysis (RTPCA) under the t-product/t-SVD framework, bypassing randomness assumptions. By introducing tangent spaces for the low-rank and sparse components and defining the rank-sparsity uncertainty principle through $\xi(fcalL)$ and $\\mu(fcalE)$, the authors derive a deterministic condition $\xi(fcalL_0)\\mu(fcalE_0)<1/6$ ensuring unique recovery of $(fcalL_0,fcalE_0)$ from $fcalX=fcalL_0+fcalE_0$ via a convex RTPCA objective with a tunable $\\gamma$. They further connect these measures to tensor incoherence and sparsity patterns, yielding practical corollaries such as bounds in terms of inc$(fcalL)$ and deg$_{min/max}(fcalE)$. The results are supported by dual certificate proofs and projections onto tangent spaces, with implications for exact recovery in moving-object tracking, image recovery, and background modeling, and opportunities for extensions to tensor completion and non-convex methods.

Abstract

Tensor, also known as multi-dimensional array, arises from many applications in signal processing, manufacturing processes, healthcare, among others. As one of the most popular methods in tensor literature, Robust tensor principal component analysis (RTPCA) is a very effective tool to extract the low rank and sparse components in tensors. In this paper, a new method to analyze RTPCA is proposed based on the recently developed tensor-tensor product and tensor singular value decomposition (t-SVD). Specifically, it aims to solve a convex optimization problem whose objective function is a weighted combination of the tensor nuclear norm and the l1-norm. In most of literature of RTPCA, the exact recovery is built on the tensor incoherence conditions and the assumption of a uniform model on the sparse support. Unlike this conventional way, in this paper, without any assumption of randomness, the exact recovery can be achieved in a completely deterministic fashion by characterizing the tensor rank-sparsity incoherence, which is an uncertainty principle between the low-rank tensor spaces and the pattern of sparse tensor.

Robust Tensor Principal Component Analysis: Exact Recovery via Deterministic Model

TL;DR

This work establishes deterministic exact recovery guarantees for Robust Tensor Principal Component Analysis (RTPCA) under the t-product/t-SVD framework, bypassing randomness assumptions. By introducing tangent spaces for the low-rank and sparse components and defining the rank-sparsity uncertainty principle through and , the authors derive a deterministic condition ensuring unique recovery of from via a convex RTPCA objective with a tunable . They further connect these measures to tensor incoherence and sparsity patterns, yielding practical corollaries such as bounds in terms of inc and deg. The results are supported by dual certificate proofs and projections onto tangent spaces, with implications for exact recovery in moving-object tracking, image recovery, and background modeling, and opportunities for extensions to tensor completion and non-convex methods.

Abstract

Tensor, also known as multi-dimensional array, arises from many applications in signal processing, manufacturing processes, healthcare, among others. As one of the most popular methods in tensor literature, Robust tensor principal component analysis (RTPCA) is a very effective tool to extract the low rank and sparse components in tensors. In this paper, a new method to analyze RTPCA is proposed based on the recently developed tensor-tensor product and tensor singular value decomposition (t-SVD). Specifically, it aims to solve a convex optimization problem whose objective function is a weighted combination of the tensor nuclear norm and the l1-norm. In most of literature of RTPCA, the exact recovery is built on the tensor incoherence conditions and the assumption of a uniform model on the sparse support. Unlike this conventional way, in this paper, without any assumption of randomness, the exact recovery can be achieved in a completely deterministic fashion by characterizing the tensor rank-sparsity incoherence, which is an uncertainty principle between the low-rank tensor spaces and the pattern of sparse tensor.

Paper Structure

This paper contains 12 sections, 10 theorems, 66 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Related Work
  3. Organizations of the Paper
  4. Notations and Preliminaries
  5. Main Results
  6. Deterministic Exact Recovery
  7. Characterization of Low-Rank and Sparse Tensors
  8. Proofs for Main Results
  9. Proofs of Lemmas \ref{['lemma: unique decomposition']} and \ref{['lemma: sufficient for trivial intersection']}
  10. Proof of Theorem \ref{['thm: exact recovery 1']}
  11. Proofs of Lemmas \ref{['lemma: low-rank tensor property']} and \ref{['lemma: sparsity pattern']}
  12. Conclusion

Key Result

Theorem 1

Let $\BFcalA\in\mathbb{R}^{N_1\times N_2\times N_3}$. Then it can factorized as where $\BFcalU \in\mathbb{R}^{N_1\times N_1\times N_3}, \BFcalV \in\mathbb{R}^{N_2\times N_2\times N_3}$ are orthogonal, and $\BFcalS\in\mathbb{R}^{N_1\times N_2\times N_3}$ is an f-diagonal tensor.

Figures (2)

  • Figure 1: Illustrations of RTPCA lu2016tensor
  • Figure 2: Illustration of t-SVD for an $N_1 \times N_2 \times N_3$ tensor liu2018improved

Theorems & Definitions (26)

  • Definition 1: T-product kilmer2011factorization
  • Definition 2: Tensor transpose kilmer2011factorization
  • Definition 3: Identity tensor kilmer2011factorization
  • Definition 4: Orthogonal tensor kilmer2011factorization
  • Definition 5: F-diagonal Tensor kilmer2011factorization
  • Theorem 1: T-SVD kilmer2011factorization
  • Definition 6: Tensor tubal rank kilmer2013thirdzhang2014novel
  • Remark 1
  • Definition 7: Tensor spectral norm lu2019tensor
  • Definition 8: Tensor nuclear norm lu2019tensor
  • ...and 16 more