Kai Zhong, Zhao Song, Prateek Jain, Inderjit S. Dhillon
Consider a standard recommendation/retrieval problem where given a query, the goal is to retrieve the most relevant items. Inductive matrix completion (IMC) method is a standard approach for this problem where the given query as well as the items are embedded in a common low-dimensional space. The inner product between a query embedding and an item embedding reflects relevance of the (query, item) pair. Non-linear IMC (NIMC) uses non-linear networks to embed the query as well as items, and is known to be highly effective for a variety of tasks, such as video recommendations for users, semantic web search, etc. Despite its wide usage, existing literature lacks rigorous understanding of NIMC models. A key challenge in analyzing such models is to deal with the non-convexity arising out of non-linear embeddings in addition to the non-convexity arising out of the low-dimensional restriction of the embedding space, which is akin to the low-rank restriction in the standard matrix completion problem. In this paper, we provide the first theoretical analysis for a simple NIMC model in the realizable setting, where the relevance score of a (query, item) pair is formulated as the inner product between their single-layer neural representations. Our results show that under mild assumptions we can recover the ground truth parameters of the NIMC model using standard (stochastic) gradient descent methods if the methods are initialized within a small distance to the optimal parameters. We show that a standard tensor method can be used to initialize the solution within the required distance to the optimal parameters. Furthermore, we show that the number of query-item relevance observations required, a key parameter in learning such models, scales nearly linearly with the input dimensionality thus matching existing results for the standard linear inductive matrix completion.