Constructive Q -Matrix Identifiability via Novel Tensor Unfolding

Abstract This work establishes a new identifiability theory for a cornerstone of various cognitive diagnostic models (CDMs) popular in psychometrics: the Q-matrix. The key idea is a novel tensor-unfolding proof strategy. Representing the joint distribution of J categorical responses as a J -way tensor, we strategically unfold the tensor into matrices in multiple ways and use their rank properties to identify the unknown Q-matrix. This approach departs fundamentally from all prior identifiability analyses in CDMs. Our proof is constructive, elucidating a population-level procedure to exactly recover the Q-matrix within a parameter space where each latent attribute is measured by at least two “pure” items that solely measure this attribute. The theory has several desirable features: it can constructively identify both the Q-matrix and the number of latent attributes; it applies to broad classes of linear and nonlinear CDMs with main or all saturated effects of attributes; and it accommodates polytomous responses, extending beyond classical binary response settings. The new identifiability result unifies and strengthens identifiability guarantees across diverse CDMs. It provides rigorous theoretical foundations and indicates a future pathway toward using tensor unfolding for practical Q-matrix estimation.