Relevance and diversity are both important to the success of recommender systems, as they help users to discover from a large pool of items a compact set of candidates that are not only interesting but exploratory as well. The challenge is that relevance and diversity usually act as two competing objectives in conventional recommender systems, which necessities the classic trade-off between exploitation and exploration. Traditionally, higher diversity often means sacrifice on relevance and vice versa. We propose a new approach, heterogeneous inference, which extends the general collaborative filtering (CF) by introducing a new way of CF inference, negative-to-positive. Heterogeneous inference achieves divergent relevance, where relevance and diversity support each other as two collaborating objectives in one recommendation model, and where recommendation diversity is an inherent outcome of the relevance inference process. Benefiting from its succinctness and flexibility, our approach is applicable to a wide range of recommendation scenarios/use-cases at various sophistication levels. Our analysis and experiments on public datasets and real-world production data show that our approach outperforms existing methods on relevance and diversity simultaneously.