On the Identifiability of Tensor Ranks via Prior Predictive Matching

TL;DR

This paper converts a set of moment matching conditions into a log-linear system of equations in terms of marginal moments, prior hyperparameters, and ranks; establishing an equivalence between rank identifiability and the solvability of such system.

Abstract

Selecting the latent dimensions (ranks) in tensor factorization is a central challenge that often relies on heuristic methods. This paper introduces a rigorous approach to determine rank identifiability in probabilistic tensor models, based on prior predictive moment matching. We transform a set of moment matching conditions into a log-linear system of equations in terms of marginal moments, prior hyperparameters, and ranks; establishing an equivalence between rank identifiability and the solvability of such system. We apply this framework to four foundational tensor-models, demonstrating that the linear structure of the PARAFAC/CP model, the chain structure of the Tensor Train model, and the closed-loop structure of the Tensor Ring model yield solvable systems, making their ranks identifiable. In contrast, we prove that the symmetric topology of the Tucker model leads to an underdetermined system, rendering the ranks unidentifiable by this method. For the identifiable models, we derive explicit closed-form rank estimators based on the moments of observed data only. We empirically validate these estimators and evaluate the robustness of the proposal.

Figures (4)

• Figure 1: Combined estimation results for the PARAFAC/CP model. The dashed line represents perfect estimation ($y=x$). Green dots show the mean of median estimates across multiple runs, with error bars indicating $\pm1$ standard deviation.
• Figure 2: Theoretical SNR (TT, Gamma prior) for $M\in\{3, 4,5\}$, $\mu=1.0$. Vertical lines mark theoretical optimal region for $\mathrm{cv}^{*}$.
• Figure 3: Tensor Train (TT) rank estimation under varying dimensions and Gamma--Poisson generative settings. Each panel reports median rank estimates (over 20 independent runs) grouped by true rank, with boxplots showing the distribution of empirical estimates. The dashed diagonal ($y{=}x$) indicates perfect recovery. All models are trained using the covariance-only TT estimator. Top row: TT tensors of size $(25,25,25)$ with Poisson likelihood and Gamma priors $\mathrm{Gamma}(\alpha,\theta)$ set to $(1.2,1.5)$ (left) and $(1.25,1.5)$ (right). Middle row: TT tensors of sizes $(24,24,24)$ and $(40,40,40)$ with $\mathrm{Gamma}(1.25,1.5)$ priors, showing improved stability with increasing tensor size. Bottom: TT tensor of size $(50,50,50)$ with $\mathrm{Gamma}(1.5,2.5)$ prior, corresponding to a higher mean and lower relative variance (lower $\mathrm{cv}$).
• Figure 4: Tensor Ring (TR) rank estimation on $(80,80,80)$ tensors with Gamma--Poisson generative models. Each panel reports median rank estimates across 20 independent runs, grouped by true rank and visualized as boxplots. The dashed diagonal ($y{=}x$) denotes perfect recovery. Both experiments use a Poisson likelihood with Gamma priors $\mathrm{Gamma}(\alpha,\theta)$ given by $(0.16,3.16)$ (left) and $(0.25,4.0)$ (right). The shape and scale values correspond to different mean–variance tradeoffs, with the higher mean/lower variance prior (right) yielding improved accuracy and reduced dispersion, consistent with the expected gain in effective signal-to-noise ratio.

Theorems & Definitions (43)

• Definition 2.1: The Probabilistic Tensor-Model
• Definition 2.2: Total Covariance and Pure Interaction Terms
• Proposition 2.3: Polynomial Structure of Rate Moments
• proof
• Definition 2.4: The Log-Linear System
• Proposition 2.5: Principle of Rank Identifiability
• Definition 3.1: The Tucker model
• Lemma 3.2: Moment Structure of the Tucker Model
• Lemma 3.3: An Identity Among Observables
• proof