低标号率下Laplacian半监督学习的收敛速度
Rates of Convergence for Laplacian Semi-Supervised Learning with Low Labeling Rates
摘要：
我们在低标记率下研究基于图的拉普拉斯半监督学习。拉普拉斯学习使用图上的调和扩展来传播标签。在非常低的标记率下，拉普拉斯学习变得退化，并且解决方案在每个标记数据点处大致恒定，具有尖峰。以前的工作表明，这种退化发生在标记数据点的数量是有限的，而未标记的数据点的数量趋于无穷大。在这项工作中，我们允许标记的数据点的数量随着标签的数量增长到无穷大。结果表明，对于长度为$\varepsilon>0$，标号率为$β>0$的随机几何图，若$β\ll\varepsilon^2$，则解退化并形成尖峰;若$β\gg \varepsilon^2$，则Laplacian学习是适定的，且与连续Laplace方程一致.此外，在适定性设置，我们证明了定量误差估计的$O(\varepsilonβ^{-1/2})$的离散问题和连续PDE的解决方案之间的差异，对数因子。我们还研究了$p$ -Laplacian正则化，并证明了当$β\ll \varepsilon^p$ .我们的适定性结果的证明使用Laplacian学习和PDE参数的随机游动解释，而不适定性结果的证明使用变分法的$Γ$收敛工具。我们还提出了模拟和真实数据的数值结果来说明我们的结果。
英文摘要：
We study graph-based Laplacian semi-supervised learning at low labeling rates. Laplacian learning uses harmonic extension on a graph to propagate labels. At very low label rates, Laplacian learning becomes degenerate and the solution is roughly constant with spikes at each labeled data point. Previous work has shown that this degeneracy occurs when the number of labeled data points is finite while the number of unlabeled data points tends to infinity. In this work we allow the number of labeled data points to grow to infinity with the number of labels. Our results show that for a random geometric graph with length scale $\varepsilon>0$ and labeling rate $β>0$, if $β\ll\varepsilon^2$ then the solution becomes degenerate and spikes form, and if $β\gg \varepsilon^2$ then Laplacian learning is well-posed and consistent with a continuum Laplace equation. Furthermore, in the well-posed setting we prove quantitative error estimates of $O(\varepsilonβ^{-1/2})$ for the difference between the solutions of the discrete problem and continuum PDE, up to logarithmic factors. We also study $p$-Laplacian regularization and show the same degeneracy result when $β\ll \varepsilon^p$. The proofs of our well-posedness results use the random walk interpretation of Laplacian learning and PDE arguments, while the proofs of the ill-posedness results use $Γ$-convergence tools from the calculus of variations. We also present numerical results on synthetic and real data to illustrate our results.
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