# On the Identifiability of Tensor Ranks via Prior Predictive Matching

arXiv: https://arxiv.org/abs/2510.14523 · HTML: https://mlsanity.com/paper/2510.14523/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 7 of its 32 references that are in the map.
- First posted on arXiv 16 Oct 2025.

## Builds on (most cited first, 7 of 7)

- [Tensorizing Neural Networks](https://mlsanity.com/paper/1509.06569/) (2015)
- [Tensor Ring Decomposition](https://mlsanity.com/paper/1606.05535/) (2016)
- [Bayesian CP Factorization of Incomplete Tensors with Automatic Rank Determination](https://mlsanity.com/paper/1401.6497/) (2014)
- [Bayesian Poisson Tensor Factorization for Inferring Multilateral Relations from Sparse Dyadic Event Counts](https://mlsanity.com/paper/1506.03493/) (2015)
- [Inference of hyperedges and overlapping communities in hypergraphs](https://mlsanity.com/paper/2204.05646/) (2022)
- [Empirical Bayes Matrix Factorization](https://mlsanity.com/paper/1802.06931/) (2018)
- [Prior Specification for Bayesian Matrix Factorization via Prior Predictive Matching](https://mlsanity.com/paper/1910.12263/) (2019)

## Abstract (arXiv)

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.