论文标题
繁殖内核产生的小波框架的构建和蒙特卡洛估计
Construction and Monte Carlo estimation of wavelet frames generated by a reproducing kernel
论文作者
论文摘要
我们在通用域中介绍了多尺度紧密框架的结构。框架元素是通过光谱滤波与重现核相关的积分算子获得的。我们的构造在连续和离散的非欧几里得结构（例如Riemannian歧管和加权图）上扩展了经典小波以及广义小波。此外，它允许在随机抽样方面研究连续和离散帧之间的关系，在这种采样方案中，可以将离散帧视为连续帧的蒙特卡洛估计值。将光谱正则化与学习理论配对，我们表明样品框架趋向于其人群对应，并在Sobolev和Besov规律性的空间上得出明确的有限样本率。我们的结果证明了基于经验数据构建的帧的稳定性，从某种意义上说，所有随机离散均具有相同的基本限制，而不论初始训练样本集如何。
We introduce a construction of multiscale tight frames on general domains. The frame elements are obtained by spectral filtering of the integral operator associated with a reproducing kernel. Our construction extends classical wavelets as well as generalized wavelets on both continuous and discrete non-Euclidean structures such as Riemannian manifolds and weighted graphs. Moreover, it allows to study the relation between continuous and discrete frames in a random sampling regime, where discrete frames can be seen as Monte Carlo estimates of the continuous ones. Pairing spectral regularization with learning theory, we show that a sample frame tends to its population counterpart, and derive explicit finite-sample rates on spaces of Sobolev and Besov regularity. Our results prove the stability of frames constructed on empirical data, in the sense that all stochastic discretizations have the same underlying limit regardless of the set of initial training samples.
