Multiscale regression on unknown manifolds
Multiscale regression on unknown manifolds
复制标题
DOI：
10.3934/mine.2022028
复制
发表时间：
2021-01
期刊：
影响因子：
--
通讯作者：
Wenjing Liao;M. Maggioni;S. Vigogna
中科院分区：
文献类型：
--
作者：
Wenjing Liao;M. Maggioni;S. Vigogna
我们考虑在$ \mathbb{R}^D $上估计函数的回归问题，但支持在$ d $维流形$ \mathcal{M} ~~\subset \mathbb{R}^D $上，其中$ d \ll D $。从多分辨率分析和非线性逼近的思想，我们构建低维坐标$ \mathcal{M} $在多个尺度，并进行多尺度回归局部多项式拟合。我们提出了一个数据驱动的小波阈值方案，自动适应未知的规律性的功能，允许有效的估计功能表现出不均匀的规律性在不同的位置和尺度。我们分析了我们的方法的推广误差，证明有限样本界的高概率丰富类的先验。我们的估计达到最佳的学习率（对数因子），就好像函数定义在一个已知的欧几里得域的维度$ d $，而不是一个未知的流形嵌入在$ \mathbb{R}^D $。所实现的算法具有准线性复杂度的样本大小，与常数线性在$ D $和指数在$ d $。因此，我们的工作建立了一个新的框架，回归低维集嵌入在高维，快速实现和强有力的理论保证。
We consider the regression problem of estimating functions on $ \mathbb{R}^D $ but supported on a $ d $-dimensional manifold $ \mathcal{M} ~~\subset \mathbb{R}^D $ with $ d \ll D $. Drawing ideas from multi-resolution analysis and nonlinear approximation, we construct low-dimensional coordinates on $ \mathcal{M} $ at multiple scales, and perform multiscale regression by local polynomial fitting. We propose a data-driven wavelet thresholding scheme that automatically adapts to the unknown regularity of the function, allowing for efficient estimation of functions exhibiting nonuniform regularity at different locations and scales. We analyze the generalization error of our method by proving finite sample bounds in high probability on rich classes of priors. Our estimator attains optimal learning rates (up to logarithmic factors) as if the function was defined on a known Euclidean domain of dimension $ d $, instead of an unknown manifold embedded in $ \mathbb{R}^D $. The implemented algorithm has quasilinear complexity in the sample size, with constants linear in $ D $ and exponential in $ d $. Our work therefore establishes a new framework for regression on low-dimensional sets embedded in high dimensions, with fast implementation and strong theoretical guarantees.