multivariance tensor decomposition

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.

This work addresses the complete (i.e., rank-$n$) decomposition of third-order symmetric tensors: given a tensor $T = sum_{i=1}^n u_i^{otimes 3}$ formed from linearly independent vectors ${u_i}_{i=1}^n subset mathbb{C}^n$, the goal is to recover ${u_i}$ up to permutation and unit-modulus phase rotations, with $ell_2$-accuracy $varepsilon$, with high probability. We propose the first randomized algorithm for finite-precision arithmetic that achieves numerically stable, high-probability recovery under the assumption that the tensor’s condition number is bounded above by $B$. The algorithm requires only $O(n^3)$ arithmetic operations and $mathrm{polylog}(n, B, 1/varepsilon)$ bits of precision. Its core innovations integrate tensor spectral analysis, condition-number-aware stability design, and explicit handling of phase invariance in complex vector recovery—thereby breaking the prior trade-off between numerical accuracy and computational complexity.

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.

This study addresses the detection of zero patterns (i.e., structural sparsity) in high-order tensors, aiming to identify generalized cluster patterns that characterize multi-parameter interactions. We propose a novel tensor clustering framework grounded in Lie algebra theory—the first to incorporate Lie group/algebra constraints into tensor structural learning—yielding a computationally tractable family of continuous cluster patterns. Our method integrates multilinear mapping contraction, direct-sum decomposition of mode-wise subspaces, and Lie-driven constrained optimization, enabling discovery of non-rigid, curve- or surface-like structures. Evaluated on synthetic and multimodal real-world data, the approach accurately recovers non-block-diagonal, non-orthogonal, and smoothly varying cluster structures. It demonstrates significantly superior structural interpretability and generalizability compared to classical tensor decomposition methods, including block-diagonal and orthogonal Tucker decompositions, as well as discrete clustering approaches.

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 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 insufficient channel estimation accuracy in MIMO systems under low signal-to-noise ratio (SNR) conditions by proposing a higher-order representation method based on modal tensorization. The approach decomposes the channel tensor into multiple virtual modes and integrates canonical polyadic decomposition with sparse structural modeling to enhance path separability and intrinsic denoising capability. Furthermore, it introduces a virtual factor analysis metric grounded in the plane-wave propagation model to enable accurate tensor rank estimation and adaptive selection of dominant components. Compared to existing tensor-based methods, the proposed scheme significantly improves channel estimation performance at low SNR, demonstrating the distinct advantage of synergistically combining modal tensorization with structural priors.

This work addresses the limitations of traditional tensor decomposition methods in effectively capturing the complex spatial-spectral coupling and cross-domain dependencies inherent in hyperspectral images. To this end, we propose a holistic multi-variate tensor decomposition framework that, for the first time, explicitly models both isolated spatial-spectral features and high-dimensional collaborative interactions while preserving essential joint multi-variate structures even under strong subspace compression. By leveraging a structurally flexible tensor algorithm, our approach overcomes the low-rank approximation constraints of conventional Tucker and CP decompositions, enabling highly discriminative feature extraction. Extensive experiments on four benchmark hyperspectral datasets demonstrate that the proposed method significantly outperforms existing decomposition techniques and consistently enhances the classification accuracy of various supervised learning algorithms.

本文介绍了一种通过计算交换算子的联合特征向量来解决多项式根和张量分解问题的方法，并分析了其多重结构。

This study addresses the identification and estimation of factor loadings in high-dimensional tensor time series. Modeling the data via CP decomposition, the authors leverage temporal dependence to construct a structured matrix and obtain an initial estimate through spectral analysis, followed by a proposed double-projection iterative algorithm to refine accuracy. The method overcomes conventional assumptions requiring factors to be approximately orthogonal or independent, accommodating correlated factors, non-orthogonal loadings, sparsity, and weak factors. Theoretically, the iterative estimator achieves a faster convergence rate and possesses an asymptotically normal distribution amenable to inference. Extensive simulations and analyses of two real-world datasets demonstrate the method’s superior effectiveness and robustness across diverse scenarios.