Alternating Recursive Method for Q-matrix Learning

The key issue affecting Cognitive Diagnostic Models (CDMs) is how to specify attributes and the Q-matrix. In this paper, we first attempt to use the Boolean Matrix Factorization (BMF) method to express conjunctive models in CDMs. Because BMF is an NPhard problem [2], we propose a recursive method that updates the attribute matrix (its rank equals to one) in each step. As Boolean algebra is irreversible, it requires time to recursively compute and update the matrix, especially when the number of attributes is large. To speed up computations, we use a Heaviside step function, which allows us to decompose the recursive computing process into normal non-negative matrices and get the results by mapping them back into a Boolean matrix. Two different algorithms are presented: a deterministic heuristic algorithm and a stochastic algorithm. Simulation results from an actual test show that the proposed method can learn the original Q-matrix well from item response data.