Tensor CUR Decomposition under the Linear-Map-Based Tensor-Tensor Multiplication

The factorization of three-dimensional data continues to gain attention due to its relevance in representing and compressing large-scale datasets. The linear-map-based tensor-tensor multiplication is a matrix-mimetic operation that extends the notion of matrix multiplication to higher order tensors, and which is a generalization of the T-product. Under this framework, we introduce the tensor CUR decomposition, show its performance in video foreground-background separation for different linear maps and compare it to a robust matrix CUR decomposition, another tensor approximation and the slice-based singular value decomposition (SS-SVD). We also provide a theoretical analysis of our tensor CUR decomposition, extending classical matrix results to establish exactness conditions and perturbation bounds.

💡 Research Summary

The paper introduces a novel tensor CUR decomposition built upon a linear‑map‑based tensor‑tensor product (denoted ∗_M), which generalizes the classical T‑product by allowing an arbitrary full‑rank linear map M∈ℝ^{q×p}. The ∗_M product is defined as
 A ∗_M B = ((A ×₃ M) △ (B ×₃ M)) ×₃ M^{-1}
where “×₃” denotes mode‑3 multiplication and “△” is a face‑wise (frontal‑slice) matrix product. Depending on whether M is invertible, surjective, or injective, the inverse is replaced by the appropriate pseudoinverse, leading to three distinct algebraic regimes.

Using this product, the authors construct a tensor CUR factorization. Given index sets I⊂{1,…,m} (rows) and J⊂{1,…,n} (columns), they define C = A(:,J,:), R = A(I,:,:), and U = A(I,J,:). The tensor A is first mapped to the M‑space (bA = A ×₃ M) and the same mapping is applied to C, R, and U, yielding bC, bR, bU. For each frontal slice k (k=1,…,q) they perform a standard matrix CUR decomposition:
 bA(k) = bC(k)·bU(k)^{+}·bR(k) + E(k)
where bU(k)^{+} is the (Moore‑Penrose) pseudoinverse. Stacking the slice‑wise factors produces bC, bU, bR, and the final approximation in the original domain is
 A ≈ C ∗_M U^{+} ∗_M R, with U^{+} = (bU)^{+} ×₃ M^{+}.

Theoretical contributions are twofold. Theorem 3.1 shows that if the multirank condition rank_m(U) = rank_m(A) holds (i.e., each frontal slice of U has the same rank as the corresponding slice of A), then for an invertible M the CUR reconstruction is exact: A = C ∗_M U^{+} ∗_M R. The proof leverages the exactness of matrix CUR slice‑wise and the linearity of the ∗_M product. Theorem 3.2 provides a perturbation bound: for a noisy tensor à = A + E with sufficiently small ∗M‑spectral norm, the CUR approximation Ā satisfies
 ‖A – Ā‖{2,∗M} ≤ C·‖E‖{2,∗_M}
where C depends only on the ∗_M‑norms of C, U^{+}, and R, mirroring known matrix CUR perturbation results.

Empirically, the method is applied to foreground‑background separation in video sequences, modeled as X = L* + S* where L* is low‑rank (multirank) and S* is sparse. The low‑rank component is approximated using the proposed ∗_M‑CUR, with column/row indices selected by the Q‑DEIM algorithm. Four choices of M are examined: (i) discrete cosine transform (DCT), (ii) discrete Fourier transform (DFT) – which reduces ∗_M to the classic T‑product, (iii) discrete sine transform (DST), and (iv) a data‑dependent matrix U₃ obtained from the SVD of the mode‑3 unfolding of the video.

Quantitative results (Table 2) report Average Gray‑level Error (AGE), Percentage of Error Pixels (pEPs), Peak Signal‑to‑Noise Ratio (PSNR), and runtime across four video clips from the SBI and CDnet datasets. In the invertible case (DCT, DFT, DST) the ∗_M‑CUR consistently achieves lower AGE and pEPs and higher PSNR than a robust matrix CUR, a tensor Bhattacharyya‑Messner (BM) decomposition, and the slice‑based SVD (SS‑SVD). DFT (i.e., the T‑product) often yields the best trade‑off between accuracy and speed. Runtime varies with the dimensionality of M: injective maps (2p×p) are the slowest, surjective maps (p×5) the fastest, while the invertible case (p×p) lies in between.

Qualitative visual comparisons (Figure 1) demonstrate that ∗_M‑CUR produces cleaner foreground masks with sharper object boundaries and fewer artifacts than the competing methods. The BM decomposition, while accurate, is 50–500 times slower, limiting its practicality for real‑time video analysis.

The paper’s contributions are: (1) a unified CUR framework for tensors under the general ∗_M product, (2) exactness and perturbation theory extending matrix CUR results to the tensor setting, and (3) extensive experimental validation showing superior compression, accuracy, and computational efficiency on realistic video data. Limitations include the requirement that M be full‑rank, reliance on slice‑wise matrix CUR (which may incur memory and communication overhead), and dependence on Q‑DEIM for index selection. Future directions suggested are extensions to non‑square or low‑rank M, higher‑order tensors, randomized CUR sampling with probabilistic guarantees, and optimized parallel implementations (GPU or distributed) to further accelerate the slice‑wise operations.