apply low-rank factorization

This work proposes a novel tensor decomposition framework, termed HMD, which addresses a key limitation of traditional low-rank methods such as Tucker and CP decompositions: their inability to capture high-order interactions among modes, as they only model mode-wise independent variations. HMD explicitly incorporates inter-modal high-order couplings within the low-rank approximation by introducing specially designed projection operators that jointly encode both isolated mode-specific effects and cross-modal interaction structures. By transcending the structural constraints inherent in conventional decompositions, the proposed method achieves substantially lower reconstruction errors and demonstrates superior fidelity and robustness across three diverse benchmark datasets, consistently outperforming Tucker and CP decompositions.

High-order tensor recovery suffers from severe non-convexity due to multiplicative coupling among factors in structural tensor models. Method: This paper proposes an orthogonal factorization framework based on Stiefel manifold optimization, bypassing conventional alternating minimization and instead performing Riemannian gradient descent directly on the manifold. It establishes, for the first time, a unified Riemannian regularity condition applicable to multiple decomposition formats—including Tucker and tensor-train (TT) decompositions. Contribution/Results: Theoretically, the algorithm converges linearly to the ground-truth tensor, with convergence rate and initialization requirements scaling only polynomially in the tensor order (N)—overcoming prior exponential dependencies. Empirically, the framework significantly improves computational scalability and recovery accuracy in high-order settings.

This paper investigates the detectability and reconstruction of low multilinear-rank signals in high-dimensional spiked tensor models near computational phase transition thresholds. To address key bottlenecks—degraded performance of conventional methods in the critical signal-to-noise ratio (SNR) regime and lack of theoretical convergence guarantees for the Higher-Order Orthogonal Iteration (HOOI) algorithm—the authors systematically apply random matrix theory to analyze the spectral properties of tensor unfoldings. They derive a novel SNR criterion characterizing statistical detectability, precisely quantify the reconstruction error of truncated multilinear singular value decomposition (MLSVD) in the nontrivial regime, and rigorously prove that, in the large-dimensional limit, HOOI converges to the global optimum in a single iteration. These results establish tight theoretical bounds for low-rank tensor estimation and provide provably efficient algorithmic guarantees.

Efficiently recovering large-scale low-tubal-rank tensors from a small number of noisy linear measurements remains challenging, as existing t-SVD–based methods suffer from high computational complexity and poor scalability. Method: This paper introduces, for the first time, a Burer–Monteiro–type bi-factorization framework into low-tubal-rank tensor recovery. We propose a Factorized Gradient Descent (FGD) algorithm that operates without prior knowledge of the true tubal rank and is robust to rank overestimation. Leveraging t-product algebra, our nonconvex optimization model avoids explicit t-SVD computation. Contribution/Results: We establish theoretical convergence guarantees under noise. Experiments on multiple benchmark tasks demonstrate that FGD achieves faster convergence, lower reconstruction error, and significantly reduced computational and storage overhead compared to state-of-the-art tensor recovery methods.

This paper addresses the robust reconstruction of undercomplete decompositions for noisy third-order symmetric tensors of rank $r leq n$: efficiently recovering linearly independent factor vectors when the input tensor is close to being decomposable. We propose the first randomized algorithm with $O(n^3)$ time complexity—matching the theoretical lower bound—and achieving inverse quasi-polynomial robustness to noise (i.e., tolerating noise magnitude up to $1/mathrm{poly}(n)$). To establish average-case efficiency and high-probability $varepsilon$-accurate recovery, we introduce a smoothed analysis framework for tensor decomposition condition numbers. Key technical innovations include implicit tensor representation, adaptive basis transformation, spectral methods, and condition-number theory. Under exact arithmetic, for tensors with $mathrm{poly}(n)$ condition number and $1/mathrm{poly}(n)$ target accuracy, the algorithm outputs $varepsilon$-approximate factor vectors with high probability.

This work addresses the Hadamard decomposition problem—recovering two low-rank matrices whose element-wise product approximates a given matrix. By reformulating the problem as a structured matrix factorization with explicit constraints, the authors propose three efficient algorithms: a direct decomposition method based on Manopt, a block projected gradient approach, and a projection-free Riemannian gradient descent scheme. A novel initialization strategy is also introduced to enhance solution accuracy. The proposed methods are particularly well-suited for large-scale sparse data and demonstrate significant improvements over truncated SVD and existing Hadamard decomposition techniques on both synthetic and real-world datasets, confirming their computational efficiency and competitive performance.

This work addresses the limitation of conventional element-wise reconstruction error in tensor low-rank approximation, which fails to capture geometric degradation of multidimensional structures. Building upon the orthogonal Tucker model, the paper introduces a novel orthogonal decomposition of reconstruction error into directional loss—quantifying subspace deviations caused by truncation and noise—and interaction loss—measuring distortions in the multilinear interactions of the core tensor. A Wedin-type stability bound is established for the directional loss. Experiments on synthetic and hyperspectral data, leveraging matrix SVD, tensor Tucker decomposition, and subspace perturbation analysis, demonstrate that under identical reconstruction errors, directional loss can vary by up to 4.6×, and high directional loss strongly correlates with visual blurring, thereby underscoring the necessity of structure-aware error metrics.

This work addresses the challenge of efficiently compressing large-scale datasets in the era of big data while supporting high-precision complex mathematical operations. Building upon the star-M tensor framework, the paper proposes a high-performance parallel tensor singular value decomposition (SVD) algorithm tailored for shared-memory architectures. It presents the first system-level language implementation of parallelized star-M SVD, overcoming the performance limitations inherent in prior approaches confined to productivity-oriented languages. By integrating tensor decomposition with advanced high-performance numerical computing techniques, the method substantially enhances both compression efficiency and reconstruction accuracy on scientific datasets, thereby establishing a robust foundation for downstream data analysis and insight extraction.