A Spectral Approach to Item Response Theory

The Rasch model is one of the most fundamental models in item response theory and has wide-ranging applications from education testing to recommendation systems. In a universe with n users and m items, the Rasch model assumes that the binary response X_{li} \in \{0,1\} of a user l with parameter \theta *_l to an item i with parameter \beta *_i (e.g., a user likes a movie, a student correctly solves a problem) is distributed as \mathbb{P}(X_{li} 1) 1/(1 \exp(-(\theta *_l - \beta *_i))) . In this paper, we propose a new item estimation algorithm for this celebrated model (i.e., to estimate \beta *). The core of our algorithm is the computation of the stationary distribution of a Markov chain defined on an item-item graph. We complement our algorithmic contributions with finite-sample error guarantees, the first of their kind in the literature, showing that our algorithm is consistent and enjoys favorable optimality properties.