论文标题
汉密尔顿 - 雅各比方程在图形上，并应用了半监督学习和数据深度
Hamilton-Jacobi equations on graphs with applications to semi-supervised learning and data depth
论文作者
论文摘要
最短的路径图距离被广泛用于数据科学和机器学习，因为它们可以近似数据歧管上的基础地质距离。但是，最短的路径距离对图表中添加损坏的边缘的高度敏感，无论是通过噪声还是对抗性扰动。在本文中，我们研究了一个称为$ p $ - eikonal方程的汉密尔顿 - 雅各比方程的家族。我们证明，$ p $ - eikonal方程$ p = 1 $是图表上可证明的可靠距离型函数，而$ p \ to \ infty $限制恢复了最短路径距离。虽然$ p $ - eikonal方程并不对应于最短路径图距离，但我们仍然表明，随机几何图上$ p $ - eikonal方程的连续限制恢复了连续体中的地理密度加权距离。我们考虑$ p $ - eikonal方程的应用在数据深度和半监督学习中，并使用连续限制来证明这两种应用程序的渐近一致性结果。最后，我们显示了在真实图像数据集上使用数据深度和半监督学习的实验结果，包括MNIST，FashionMnist和CIFAR-10，这表明与最短路径距离相比，$ P $ -Eikonal方程提供了明显更好的结果。
Shortest path graph distances are widely used in data science and machine learning, since they can approximate the underlying geodesic distance on the data manifold. However, the shortest path distance is highly sensitive to the addition of corrupted edges in the graph, either through noise or an adversarial perturbation. In this paper we study a family of Hamilton-Jacobi equations on graphs that we call the $p$-eikonal equation. We show that the $p$-eikonal equation with $p=1$ is a provably robust distance-type function on a graph, and the $p\to \infty$ limit recovers shortest path distances. While the $p$-eikonal equation does not correspond to a shortest-path graph distance, we nonetheless show that the continuum limit of the $p$-eikonal equation on a random geometric graph recovers a geodesic density weighted distance in the continuum. We consider applications of the $p$-eikonal equation to data depth and semi-supervised learning, and use the continuum limit to prove asymptotic consistency results for both applications. Finally, we show the results of experiments with data depth and semi-supervised learning on real image datasets, including MNIST, FashionMNIST and CIFAR-10, which show that the $p$-eikonal equation offers significantly better results compared to shortest path distances.
