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Multi-vector retrieval models improve over single-vector dual encoders on many information retrieval tasks. In this paper, we cast the multi-vector retrieval problem as sparse alignment between query and document tokens. We propose AligneR, a novel multi-vector retrieval model that learns sparsified pairwise alignments between query and document tokens (e.g. `dog' vs. `puppy') and per-token unary saliences reflecting their relative importance for retrieval. We show that controlling the sparsity of pairwise token alignments often brings significant performance gains. While most factoid questions focusing on a specific part of a document require a smaller number of alignments, others requiring a broader understanding of a document favor a larger number of alignments. Unary saliences, on the other hand, decide whether a token ever needs to be aligned with others for retrieval (e.g. `kind' from `kind of currency is used in new zealand}'). With sparsified unary saliences, we are able to prune a large number of query and document token vectors and improve the efficiency of multi-vector retrieval. We learn the sparse unary saliences with entropy-regularized linear programming, which outperforms other methods to achieve sparsity. In a zero-shot setting, AligneR scores 51.1 points nDCG@10, achieving a new retriever-only state-of-the-art on 13 tasks in the BEIR benchmark. In addition, adapting pairwise alignments with a few examples (<= 8) further improves the performance up to 15.7 points nDCG@10 for argument retrieval tasks. The unary saliences of AligneR helps us to keep only 20% of the document token representations with minimal performance loss. We further show that our model often produces interpretable alignments and significantly improves its performance when initialized from larger language models.