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Keyphrase extraction is a fundamental task in natural language processing and information retrieval that aims to extract a set of phrases with important information from a source document. Identifying important keyphrase is the central component of the keyphrase extraction task, and its main challenge is how to represent information comprehensively and discriminate importance accurately. In this paper, to address these issues, we design a new hyperbolic matching model (HyperMatch) to represent phrases and documents in the same hyperbolic space and explicitly estimate the phrase-document relevance via the Poincaré distance as the important score of each phrase. Specifically, to capture the hierarchical syntactic and semantic structure information, HyperMatch takes advantage of the hidden representations in multiple layers of RoBERTa and integrates them as the word embeddings via an adaptive mixing layer. Meanwhile, considering the hierarchical structure hidden in the document, HyperMatch embeds both phrases and documents in the same hyperbolic space via a hyperbolic phrase encoder and a hyperbolic document encoder. This strategy can further enhance the estimation of phrase-document relevance due to the good properties of hyperbolic space. In this setting, the keyphrase extraction can be taken as a matching problem and effectively implemented by minimizing a hyperbolic margin-based triplet loss. Extensive experiments are conducted on six benchmarks and demonstrate that HyperMatch outperforms the state-of-the-art baselines.