Using mixture models for collaborative filtering

A collaborative filtering system at an e-commerce site or similar service uses data about aggregate user behavior to make recommendations tailored to specific user interests. We develop recommendation algorithms with provable performance guarantees in a probabilistic mixture model for collaborative filtering proposed by Hofmann and Puzicha. We identify certain novel parameters of mixture models that are closely connected with the best achievable performance of a recommendation algorithm; we show that for any system in which these parameters are bounded, it is possible to give recommendations whose quality converges to optimal as the amount of data grows. All our bounds depend on a new measure of independence that can be viewed as an L"1-analogue of the smallest singular value of a matrix. Using this, we introduce a technique based on generalized pseudoinverse matrices and linear programming for handling sets of high-dimensional vectors. We also show that standard approaches based on L"2 spectral methods are not strong enough to yield comparable results, thereby suggesting some inherent limitations of spectral analysis.