Title: Matrix Completion via Factorizing Polynomials,
Abstract: Predicting unobserved entries of a partially observed matrix has found wide
applicability in several areas, such as recommender systems, computational
biology, and computer vision. Many scalable methods with rigorous theoretical
guarantees have been developed for algorithms where the matrix is factored into
low-rank components, and embeddings are learned for the row and column
entities. While there has been recent research on incorporating explicit side
information in the low-rank matrix factorization setting, often implicit
information can be gleaned from the data, via higher-order interactions among
entities. Such implicit information is especially useful in cases where the
data is very sparse, as is often the case in real-world datasets. In this
paper, we design a method to learn embeddings in the context of recommendation
systems, using the observation that higher powers of a graph transition
probability matrix encode the probability that a random walker will hit that
node in a given number of steps. We develop a coordinate descent algorithm to
solve the resulting optimization, that makes explicit computation of the higher
order powers of the matrix redundant, preserving sparsity and making
computations efficient. Experiments on several datasets show that our method,
that can use higher order information, outperforms methods that only use
explicitly available side information, those that use only second-order
implicit information and in some cases, methods based on deep neural networks
as well.