A while back, we wondered If Education [was] a low Rank problem or probably how to use the fact Aha moments are sparse. How about using the fact that most key concepts are really sparse ? Looks like a probabilistic matrix factorization problem:
Armed with this model and given incomplete observations of the graded learner–question responses Yi;j, our goal is to estimate the factors W, C, and M. Such a factor-analysis problem is ill-posed in general, especially when each learner answers only a small subset of the collection of questions (see Harman (1976) for a factor analysis overview). Our ﬁrst key observation that enables a well-posed solution is the fact that typical educational domains of interest involve only a small number of key concepts (i.e., we have K N; Q in Figure 1).
We develop a new model and algorithms for machine learning-based learning analytics, which estimate a learner’s knowledge of the concepts underlying a domain, and content analytics, which estimate the relationships among a collection of questions and those concepts. Our model represents the probability that a learner provides the correct response to a question in terms of three factors: their understanding of a set of underlying concepts, the concepts involved in each question, and each question’s intrinsic diﬃculty. We estimate these factors given the graded responses to a collection of questions. The underlying estimation problem is ill-posed in general, especially when only a subset of the questions are answered. The key observation that enables a well-posed solution is the fact that typical educational domains of interest involve only a small number of key concepts. Leveraging this observation, we develop both a bi-convex maximum-likelihood-based solution and a Bayesian solution to the resulting SPARse Factor Analysis (SPARFA) problem. We also incorporate user-deﬁned tags on questions to facilitate the interpretability of the estimated factors. Experiments with synthetic and real-world data demonstrate the eﬃcacy of our approach. Finally, we make a connection between SPARFA and noisy, binary-valued (1-bit) dictionary learning that is of independent interest.
Liked this entry ? subscribe to Nuit Blanche's feed, there's more where that came from. You can also subscribe to Nuit Blanche by Email, explore the Big Picture in Compressive Sensing or the Matrix Factorization Jungle and join the conversations on compressive sensing, advanced matrix factorization and calibration issues on Linkedin.