Title: Long Document Summarization with Top-Down and Bottom-Up Representation Inference. Abstract: Text summarization aims to condense long documents and retain key information. Critical to the success of a summarization model is the faithful inference of latent representations of words or tokens in the source documents. Most recent models infer the latent representations with a transformer encoder, which is purely bottom-up. Also, self-attention-based inference models face the challenge of quadratic complexity with respect to sequence length. We propose a principled inference framework to improve summarization models on these two aspects. Our framework assumes a hierarchical latent structure of a document where the top-level captures the long range dependency at a coarser time scale and the bottom token level preserves the details. Critically, this hierarchical structure enables token representations to be updated in both a bottom-up and top-down manner. In the bottom-up pass, token representations are inferred with local self-attention to leverage its efficiency. Top-down correction is then applied to allow tokens to capture long-range dependency. We demonstrate the effectiveness of the proposed framework on a diverse set of summarization datasets, including narrative, conversational, scientific documents and news. Our model achieves (1) competitive or better performance on short documents with higher memory and compute efficiency, compared to full attention transformers, and (2) state-of--the-art performance on a wide range of long document summarization benchmarks, compared to recent efficient transformers. We also show that our model can summarize an entire book and achieve competitive performance using $0.27\%$ parameters (464M vs. 175B) and much less training data, compared to a recent GPT-3-based model. These results indicate the general applicability and benefits of the proposed framework. 