Local Popularity Based Collaborative Filters

Motivated by applications such as recommendation systems, we consider the estimation of a binary random field X obtained by unknown row and column permutations of a block constant random matrix. The estimation of X is based on observations Y, which are obtained by passing entries of X through a binary symmetric channel (BSC) (representing noisy user behavior) and an erasure channel (representing missing data). We analyze an estimation algorithm based on local popularity. We study the bit error rate (BER) in the limit as the matrix size approaches infinity and the erasure rate approaches unity at a specified rate. Our main result identifies three regimes characterized by the cluster size and erasure rate. In one regime, the algorithm has asymptotically zero BER, in another regime the BER is bounded away from 0 and 1/2, while in the remaining regime, the algorithm fails and BER approaches 1/2. Numerical results for the Movielens dataset and comparison with earlier work is also given.