3-D point clouds facilitate 3-D visual applications with detailed information of objects and scenes but bring about enormous challenges to design efficient compression technologies. The irregular signal statistics and high-order geometric structures of 3-D point clouds cannot be fully exploited by existing sparse representation and deep learning based point cloud attribute compression schemes and graph dictionary learning paradigms. In this paper, we propose a novel $p$ -Laplacian embedding graph dictionary learning framework that jointly exploits the varying signal statistics and high-order geometric structures for 3-D point cloud attribute compression. The proposed framework formulates a nonconvex minimization constrained by $p$ -Laplacian embedding regularization to learn a graph dictionary varying smoothly along the high-order geometric structures. An efficient alternating optimization paradigm is developed by harnessing ADMM to solve the nonconvex minimization. To our best knowledge, this paper proposes the first graph dictionary learning framework for point cloud compression. Furthermore, we devise an efficient layered compression scheme that integrates the proposed framework to exploit the correlations of 3-D point clouds in a structured fashion. Experimental results demonstrate that the proposed framework is superior to state-of-the-art transform-based methods in $M$ -term approximation and point cloud attribute compression and outperforms recent MPEG G-PCC reference software.