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In this paper, we propose a method, based on graph signal processing, tooptimize the choice of $k$ in $k$-nearest neighbor graphs ($k$NNGs). $k$NN isone of the most popular approaches and is widely used in machine learning andsignal processing. The parameter $k$ represents the number of neighbors thatare connected to the target node; however, its appropriate selection is still achallenging 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 foreach 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 existinggraph learning methods. In experiments on real datasets, we demonstrate thatthe $k$NNGs obtained with our method are sparse and can determine anappropriate variable number of edges per node. We validate the effectiveness ofthe proposed method for point cloud denoising, comparing our denoisingperformance with achievable graph construction methods that can be scaled totypical point cloud sizes (e.g., thousands of nodes).

Optimizing $k$ in $k$NN Graphs with Graph Learning Perspective