Nonlinear reformulations of the spectral clustering method have gained a lot of recent attention due to their increased numerical benefits and their solid mathematical background. We present a novel direct multiway spectral clustering algorithm in the $p$-norm, for $p \in (1, 2]$. The problem of computing multiple eigenvectors of the graph $p$-Laplacian, a nonlinear generalization of the standard graph Laplacian, is recasted as an unconstrained minimization problem on a Grassmann manifold. The value of $p$ is reduced in a pseudocontinuous manner, promoting sparser solution vectors that correspond to optimal graph cuts as $p$ approaches one. Monitoring the monotonic decrease of the balanced graph cuts guarantees that we obtain the best available solution from the $p$-levels considered. We demonstrate the effectiveness and accuracy of our algorithm in various artificial test-cases. Our numerical examples and comparative results with various state-of-the-art clustering methods indicate that the proposed method obtains high quality clusters both in terms of balanced graph cut metrics and in terms of the accuracy of the labelling assignment. Furthermore, we conduct studies for the classification of facial images and handwritten characters to demonstrate the applicability in real-world datasets.
翻译:光谱集群方法的非线性重制方法因其增加的数字效益和坚实的数学背景而最近引起许多关注。 我们展示了一个新的直接的多路光谱集成算法, 以美元/ 诺尔姆为单位, 以美元/ 美元(1, 2) 美元为单位, 以美元/ 美元为单位, 以美元/ 美元/ 美元为单位, 以美元/ 美元/ 美元/ 美元为单位, 以新颖的美元/ 诺尔姆为单位, 以新的直接的多路光谱集成算法。 计算“ $/ p美元- Laplacian ” 图形的多重精度, 以非线性概观性的方式将Laplacecian 方法重新表述为格拉斯曼方块上不受限制的最小化问题。 $/ plapcian 标准图的数值和比较结果显示, 以假相调的方式减少了美元, 以假相调的方式减少了美元/,,, 推广与最佳的平方位缩图削减相对的矢量的矢量矢量矢量矢量值矢量矢量值矢量,,,,,,, 以 和 以真实性 表示 的 的 的 显示 格式的 的 的 格式的 的 格式的 的 的 的 的 的 标定的 。