In this paper, we propose a method, based on graph signal processing, to optimize the choice of $k$ in $k$-nearest neighbor graphs ($k$NNGs). $k$NN is one of the most popular approaches and is widely used in machine learning and signal processing. The parameter $k$ represents the number of neighbors that are connected to the target node; however, its appropriate selection is still a challenging problem. Therefore, most $k$NNGs use ad hoc selection methods for $k$. In the proposed method, we assume that a different $k$ can be chosen for each node. We formulate a discrete optimization problem to seek the best $k$ with a constraint on the sum of distances of the connected nodes. The optimal $k$ values are efficiently obtained without solving a complex optimization. Furthermore, we reveal that the proposed method is closely related to existing graph learning methods. In experiments on real datasets, we demonstrate that the $k$NNGs obtained with our method are sparse and can determine an appropriate variable number of edges per node. We validate the effectiveness of the proposed method for point cloud denoising, comparing our denoising performance with achievable graph construction methods that can be scaled to typical point cloud sizes (e.g., thousands of nodes).

翻译：本文提出一种基于图信号处理的方法，用于优化$k$近邻图（$k$NNGs）中参数$k$的选择。$k$NN是最流行的方法之一，广泛应用于机器学习与信号处理领域。参数$k$表示与目标节点相连的邻居数目，但其合理选取仍具挑战性。因此，大多数$k$NNG采用启发式方法选择$k$。在所提方法中，我们假设每个节点可选择不同的$k$值。通过构建离散优化问题，在连接节点距离之和的约束条件下寻求最优$k$值。该方法无需求解复杂优化问题即可高效获得最优$k$值。此外，我们揭示了所提方法与现有图学习方法的紧密关联。在真实数据集实验中，证明采用本文方法获得的$k$NNG具备稀疏性，且能确定每个节点恰当的可变边数。通过点云去噪实验验证了方法的有效性，将我们的去噪性能与可扩展至典型点云规模（如数千节点）的可行图构建方法进行了比较。