Learning adaptive multiscale approximations to data and functions near low-dimensional sets
Learning adaptive multiscale approximations to data and functions near low-dimensional sets
复制标题
学习低维集附近数据和函数的自适应多尺度近似
DOI：
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发表时间：
2016
期刊：
影响因子：
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通讯作者：
S. Vigogna
中科院分区：
文献类型：
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作者：
Wenjing Liao;M. Maggioni;S. Vigogna
在ℝD中的数据集由集中在未知d维集M上或附近的概率度量ρ的样本组成的情况下，D大但d≪D，我们考虑两组问题：M的几何逼近和函数f在M上的回归。在第一种情况下，我们构造了M的多尺度低维经验逼近，当M具有可能在不同位置和尺度上变化的几何正则性时，这些经验逼近是自适应的，并给出了性能保证。在第二种情况下，我们利用这些经验几何近似来构造对M上的f的多尺度逼近，即使当f在不同的尺度和位置变化时，它也适应f的未知正则性。我们证明了当f被定义在d维的欧几里得区域上，而不是定义在未知流形M上时，我们可以获得相同的学习率。所有算法的复杂度都是O(Nlogn)，常数在D中线性增长，在d中指数增长。
In the setting where a data set in ℝD consists of samples from a probability measure ρ concentrated on or near an unknown d-dimensional set M, with D large but d ≪ D, we consider two sets of problems: geometric approximation of M and regression of a function f on M. In the first case we construct multiscale low-dimensional empirical approximations of M, which are adaptive when M has geometric regularity that may vary at different locations and scales, and give performance guarantees. In the second case we exploit these empirical geometric approximations to construct multiscale approximations to f on M, which adapt to the unknown regularity of f even when this varies at different scales and locations. We prove guarantees showing that we attain the same learning rates as if f was defined on a Euclidean domain of dimension d, instead of an unknown manifold M. All algorithms have complexity O(n log n), with constants scaling linearly in D and exponentially in d.