cp decomposition

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.

This paper addresses two fundamental limitations in CANDECOMP/PARAFAC (CP) tensor decomposition: the lack of statistical optimality guarantees under noise, non-orthogonality, and high rank, and the weak convergence theory for the alternating least squares (ALS) algorithm. To resolve these, we propose a joint framework integrating the Tensor Alternating Spectral Decomposition (TASD) initialization with ALS. We establish, for the first time, non-asymptotic minimax-optimal error bounds for CP decomposition under general order, dimension, and rank. We further characterize ALS’s two-stage convergence dynamics: quadratic convergence initially, followed by linear refinement; in the rank-one case, only 1–2 iterations suffice to achieve statistical optimality. Experiments demonstrate that TASD+ALS significantly improves estimation stability and accuracy in noisy settings. Our core contribution is a unified theoretical characterization of both the statistical limits and algorithmic convergence rates of CP decomposition, thereby filling a critical gap in high-dimensional tensor decomposition theory.

Existing PARAFAC2-CMTF models suffer from limited regularization flexibility and inflexible cross-dataset coupling mechanisms. To address these limitations, this paper proposes the first unified PARAFAC2-CMTF framework supporting full-mode constraints and linear couplings—namely, matrix-, CP-, and PARAFAC2-level couplings. Methodologically, the framework integrates alternating optimization (AO) with the alternating direction method of multipliers (ADMM), enabling non-rigid tensor modeling while accommodating diverse factor priors—including sparsity, smoothness, and non-negativity. It naturally handles irregular time-series data and heterogeneous, dynamic multi-source datasets. Extensive experiments on both synthetic and real-world benchmarks demonstrate that the proposed framework achieves significantly higher decomposition accuracy and computational efficiency than state-of-the-art methods. These results validate its generality, robustness, and practical applicability in complex, real-world tensor analytics scenarios.

This paper addresses estimation and inference for high-dimensional tensor factor models, where all tensor dimensions diverge, under a CP-type decomposition allowing non-orthogonal loadings. We propose an iterative synchronized projection estimator and, for the first time under weak dependence conditions, establish its consistency and asymptotic normality. Within a unified framework, we further develop two eigenvalue-ratio criteria for consistent identification of the number of factors. Our approach relaxes the conventional orthogonality assumption on loadings, enhancing applicability to multidimensional heterogeneous time series in economics and finance—such as international trade flows and asset return matrices. Monte Carlo simulations and empirical analysis—specifically portfolio sorting—demonstrate that the proposed method substantially outperforms existing tensor factor estimators in finite samples.

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 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 limitation of existing generalized CANDECOMP/PARAFAC (CP) decomposition methods, which typically overlook the prevalent symmetry structures in tensors—such as those arising in dynamic graphs—and thus struggle to model symmetric data effectively. To overcome this, we propose Symmetric Generalized CP (SymGCP) decomposition, which, for the first time, incorporates symmetry constraints over arbitrary subsets of tensor modes into the generalized CP framework. While preserving the flexibility of general loss functions, SymGCP derives gradient expressions compatible with existing tensor kernels and introduces an efficient stochastic optimization algorithm. Experimental results demonstrate that SymGCP significantly improves both modeling accuracy and scalability on synthetic and real-world large-scale symmetric tensor data.

This paper addresses the challenge of state-switching dynamics in latent factors of tensor time series. We propose the Threshold Tensor Factor Model (TTFM), which integrates a threshold autoregressive structure into the CANDECOMP/PARAFAC (CP) decomposition framework—enabling low-rank tensor representations to capture nonlinear regime shifts while preserving interpretability. Methodologically, TTFM jointly estimates factor loadings, latent factor trajectories, and threshold parameters via iterative optimization and statistical inference. We establish asymptotic consistency and convergence rates for all parameter estimators. Simulation studies and empirical analysis on real-world data demonstrate that TTFM significantly outperforms conventional linear tensor factor models in both in-sample fit and out-of-sample rolling forecasting accuracy. By unifying nonlinear dynamics with interpretable tensor decomposition, TTFM offers a novel paradigm for nonlinear dimensionality reduction and dynamic modeling of high-dimensional time series.

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 memory-bandwidth-bound nature of the sparse MTTKRP (spMTTKRP) operation in sparse tensor decomposition, which suffers from low efficiency on general-purpose processors. To overcome this limitation, the study introduces processing-in-memory (PIM) technology to accelerate spMTTKRP for the first time, leveraging the UPMEM PIM architecture. The authors devise an efficient sparse tensor tiling strategy, a customized numerical format, and specialized compute kernels tailored to the PIM platform, along with a CPU-PIM heterogeneous collaboration mechanism. Experimental results demonstrate that the pure PIM implementation achieves a 2.37× speedup over the state-of-the-art CPU baseline, while the heterogeneous approach yields a 2.64× improvement, both exhibiting substantially higher resource utilization efficiency compared to conventional CPU and GPU implementations.