Topologically penalized regression on manifolds
Topologically penalized regression on manifolds
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发表时间：
2021-10
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通讯作者：
Olympio Hacquard;K. Balasubramanian;G. Blanchard;W. Polonik;Clément Levrard
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作者：
Olympio Hacquard;K. Balasubramanian;G. Blanchard;W. Polonik;Clément Levrard
研究了紧致流形M上的一个回归问题。为了利用数据的基本几何和拓扑，回归任务是在流形的Laplace-Beltrami算子的前几个本征函数的基础上执行的，这些本征函数是用拓扑惩罚正则化的。所提出的惩罚是基于子水平集的本征函数或估计函数的拓扑结构。整体的方法是产生有前途的和有竞争力的性能在各种应用程序的合成和真实的数据集。我们还提供了回归函数估计的理论保证，其预测误差和平滑度（在拓扑意义上）。总之，这些结果支持我们的方法的相关性的情况下，目标函数是“拓扑光滑”。
We study a regression problem on a compact manifold M. In order to take advantage of the underlying geometry and topology of the data, the regression task is performed on the basis of the first several eigenfunctions of the Laplace-Beltrami operator of the manifold, that are regularized with topological penalties. The proposed penalties are based on the topology of the sub-level sets of either the eigenfunctions or the estimated function. The overall approach is shown to yield promising and competitive performance on various applications to both synthetic and real data sets. We also provide theoretical guarantees on the regression function estimates, on both its prediction error and its smoothness (in a topological sense). Taken together, these results support the relevance of our approach in the case where the targeted function is ''topologically smooth''.