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Finding dense semantic correspondence is a fundamental problem in computer vision, which remains challenging in complex scenes due to background clutter, extreme intra-class variation, and a severe lack of ground truth. In this paper, we aim to address the challenge of label sparsity in semantic correspondence by enriching supervision signals from sparse keypoint annotations. To this end, we first propose a teacher-student learning paradigm for generating dense pseudo-labels and then develop two novel strategies for denoising pseudo-labels. In particular, we use spatial priors around the sparse annotations to suppress the noisy pseudo-labels. In addition, we introduce a loss-driven dynamic label selection strategy for label denoising. We instantiate our paradigm with two variants of learning strategies: a single offline teacher setting, and mutual online teachers setting. Our approach achieves notable improvements on three challenging benchmarks for semantic correspondence and establishes the new state-of-the-art. Project page: https://shuaiyihuang.github.io/publications/SCorrSAN.