Tensor Decomposition-Based Inference

• Tensor decomposition-based inference is a framework that factorizes multiway data into low-rank tensors using methods like CP and Tucker, enhancing identifiability and scalability.
• Bayesian and variational approaches improve robustness by integrating priors, adaptive rank selection, and neural autoencoders in modeling latent structures.
• Algorithms such as ALS, AMP, and spectral shrinkage offer efficient, scalable factor recovery and denoising, addressing challenges in noise and missing data.

Tensor decomposition-based inference refers to statistical and algorithmic methodologies that recover, estimate, or interpret latent structure from multiway data using low-rank tensor models. These approaches generalize matrix factorization and latent variable inference by leveraging the inherent multi-dimensionality of tensors, providing improved scalability, identifiability, and robustness across a range of noise models and data types. The principal frameworks include canonical polyadic (CP) and Tucker decomposition, nonparametric Bayesian tensor processes, neural network-driven variational inference, and recent advances in approximate message passing and denoising for network and signal recovery.

1. Canonical Polyadic and Tucker Decomposition-Based Inference

Tensor decomposition models posit that a $K$-way observed tensor $$\mathcal{Y}\in\mathbb{R}^{d_1\times\dots\times d_K}$$ admits a factorization of the form: $$\mathcal{Y} \approx \sum_{r=1}^R \lambda_r\,a_r^{(1)}\!\otimes a_r^{(2)}\!\otimes\cdots\otimes a_r^{(K)}$$ where $R$ is the CP rank, $\lambda_r$ are scaling parameters, and $a_r^{(k)}$ are mode-$k$ loading vectors. Classical inference of the factors and rank proceeds by minimizing residual norms (Frobenius, nuclear) or by maximizing likelihood under noise models (e.g., Gaussian, Poisson) subject to structural or regularization constraints.

The identifiability of CP decomposition—crucial for meaningful parameter recovery—is governed by Kruskal's rank condition: for a $K$-way tensor, uniqueness up to scaling/permutation is guaranteed if $\sum_k \text{kr}(A^{(k)}) \geq 2R + (K-1)$.

Tucker decomposition generalizes CP by introducing a finite core tensor $G$ and factors: $$\mathcal{Y} \approx G \times_1 U^{(1)} \times_2 \cdots \times_K U^{(K)}$$ with mode-wise multiplication. Tucker is less constrained but generally less identifiable.

2. Bayesian, Variational, and Nonparametric Tensor Models

Statistical inference is often cast in Bayesian form, with priors over factor matrices and marginalized (integrated) likelihood. The Infinite Tucker model (Xu et al., 2011) introduces a nonparametric Bayesian approach by placing tensor-variate Gaussian or Student's $t$ process priors over the latent tensor $M$. The generative model is:

• $Y$ observed via $p(Y|M)$ (Gaussian, probit, etc)
• $M$ constructed as $$M = W \times_1 \phi(U^{(1)}) \cdots \times_K \phi(U^{(K)})$$ with $W$ infinite-dimensional and $\phi$ a kernel-induced feature map
• $W$ follows a (tensor-)normal distribution, inducing multiway covariance $$\text{cov}(m_i,m_j)=\prod_k \Sigma^{(k)}(u^{(k)}_i, u^{(k)}_j)$$

Inference employs variational EM, approximating posterior distributions $q(M)$, $q(U^{(k)})$ with tractable updates exploiting Kronecker structure for scalability.

The variational auto-encoder CP model (VAECP) (Liu et al., 2016) replaces multilinear inner products with a neural decoder $f(u^i;\theta)$, mapping latent factors $u^i$ to mean/variance of $p(x_i|u^i;\theta)$, with encoder $q(u^i|x_i;\phi)$ parameterized as a Gaussian. The evidence lower bound (ELBO) per observed entry is: $$\mathcal{L}(\theta, \phi) = \sum_{i} I_i\, \mathbb{E}_{q(u^i|x_i;\phi)}[\log p(x_i|u^i;\theta)] - \sum_{d=1}^D \sum_{i_d=1}^{N_d} \mathrm{KL}(q(U^d_{i_d:}|\phi)\| p(U^d_{i_d:}))$$ Optimization leverages the reparameterization trick and stochastic gradients (Adam), handling missing data by ignoring summation over unobserved entries.

Multiplicative shrinkage priors for CP margins, adaptive rank selection, and interweaving strategies are foundational for Bayesian tensor VARs (Luo et al., 2022):

• Margins $$\beta^{(r)}_{j,i} \sim N(0, (\phi^{-1}_{r,j,i}\tau^{-1}_r))$$
• Rank adapted by monitoring inactive margin columns
• Interweaving (ASIS) improves MCMC mixing by reparameterizing the CP factors

3. Algorithms: Alternating Least Squares, AMP, and Spectral Shrinkage

The Alternating Linear Scheme (ALS) and Bayesian ALS (Menzen et al., 2020) are block-coordinate approaches for low-rank tensor recovery, applicable in CP, Tucker, and tensor train (TT) forms. Bayesian ALS updates each core $G_n$'s posterior: $$P_n^+ = \left[(P_n^0)^{-1} + (U_{\setminus n}^T U_{\setminus n})/\sigma^2\right]^{-1}, \quad m_n^+ = P_n^+ \left(U_{\setminus n}^T y / \sigma^2 + (P_n^0)^{-1} m_n^0\right)$$ Global uncertainty in the reconstructed tensor is captured via the unscented transform in TT-format.

Tensor Generalized Approximate Message Passing (TeG-AMP) (Li et al., 25 Mar 2025) approximates BP on tensor-ring (TR) or CP graphs by propagating means and variances using local CLT and Taylor expansions. Update steps involve pseudo-data denoising, Onsager corrections, and local posterior means/variances for core variables, yielding scalable inference for noisy or incomplete tensors.

Spectral denoising and shrinkage, as in multiplex Kronecker graph inference (Khobizy, 27 Jun 2025), leverages eigenvalue thresholding and reconstructs signal by SVD/post-shrinkage (Gavish–Donoho law), with Einstein summation kernelization for dramatic reduction in computational complexity.

4. Noise Models, Efficiency, and Information-Theoretic Guarantees

Different decomposition-inference frameworks handle noise and statistical efficiency via tailored likelihoods and regularization:

• Poisson models for count tensors (López et al., 7 Nov 2025): Shifted-log-likelihood, CP-rank constraints, Fisher information and CRLB bounds for factor/parameter estimation

$$\mathcal{L}(\mathscr{X}; \Theta) = \sum (x_i + 1) \log(\theta_i + 1) - (\theta_i + 1)$$

Near-efficiency achieved in rank-1; minimax rates in higher rank.

• Bounds on Bayesian generalization error are determined by the real log canonical threshold (RLCT) (Yoshida et al., 2023), providing asymptotic $O(1/n)$ error rates and supporting Bayesian model selection via penalized BIC.

Constraint handling, identifiability, and variance minimization are crucial in signal separation problems (e.g., TBM massive access (Decurninge et al., 2021)) and moment tensor-based latent variable models (e.g., online CP for topic/community modeling (Huang et al., 2013)).

5. Applications: Network, Graphical Models, Dimensionality Reduction, and Attention

Tensor decomposition-based inference finds application in:

• Network topology inference (batch and adaptive PARAFAC on covariances (Shen et al., 2016)), exploiting uniqueness for directed graph discovery under piecewise-stationary input statistics.
• Anomaly detection (Streit et al., 2020): CP/PARAFAC decomposition defines normal subspaces; anomalies identified via residuals, with online PWO for adaptive tracking.
• Dimensionality reduction (Ju et al., 2017): Probabilistic CP-projection bases, variational EM, and feature-rank selection frameworks outperform PCA/Tucker in classification/clustering.
• Attention modules for SNNs (Deng et al., 2023): CP decomposition enables projected full attention (PFA) modules with linear parameter scaling, supporting low-rank, efficient inference and state-of-the-art classification performance.

6. Empirical and Theoretical Insights

Across simulation studies and empirical analyses:

• Variational auto-encoder CPs and infinite Tucker models outperform classical multilinear models, particularly in nonlinear or missing-value regimes (Liu et al., 2016, Xu et al., 2011).
• State-evolution analysis of TeG-AMP/TeS-AMP suggests near-MMSE performance and robustness to noise/sample sparsity (Li et al., 25 Mar 2025).
• Bayesian inference with RLCT-informed complexity control yields sharper model selection and improved generalization (Yoshida et al., 2023).
• CP-rank selection via eigenvalue-ratio estimators is consistent under high-dimensional scaling, and simultaneous orthogonalization methods provide fast, robust factor recovery for tensor-variate time series (Chen et al., 25 Jun 2024).

7. Limitations and Future Directions

Main limitations include computational bottlenecks for high CP/Tucker ranks, degeneracy or non-identifiability in factor recovery, and model selection challenges for heterogeneous noise or nonstandard marginal distributions. Promising directions include extension to arbitrary noise models (e.g. negative-binomial (López et al., 7 Nov 2025, Yoshida et al., 2023)), design of inference algorithms that approach CRLB for $R>1$, distributed or asynchronous AMP for tensors, and development of convex relaxations or robust/sparse outlier models.

In summary, tensor decomposition-based inference frameworks encompass a diversity of statistical methodologies, from multilinear algebraic approaches to fully Bayesian infinite-dimensional models and neural autoencoders, supplying theoretically grounded, scalable solutions for multiway data analysis, model selection, uncertainty quantification, and domain adaptation across increasingly complex real-world datasets.