论文标题
在高维度中恢复表面和功能：采样理论和与神经网络的联系
Recovery of surfaces and functions in high dimensions: sampling theory and links to neural networks
论文作者
论文摘要
可以将几种成像算法（包括基于斑块的图像降解，图像时间序列恢复和卷积神经网络）视为利用信号的歧管结构的方法。尽管这些算法的经验性表现令人印象深刻，但对恢复在多种多样的信号和功能的恢复的理解却较少了解。在本文中，我们专注于恢复在表面结合的信号的恢复。特别是，我们考虑在高维度的平滑带限制表面的结合中生活的信号。我们表明，指数映射将数据转换为低维子空间的结合。使用这种关系，我们引入了一个采样理论框架，以从几个样本中恢复平滑表面以及学习平滑表面上的功能。特征的低级别特性用于确定恢复表面所需的测量数量。此外，功能的低排名属性还提供了一种有效的方法，类似于神经网络，用于表面上多维函数的局部表示。
Several imaging algorithms including patch-based image denoising, image time series recovery, and convolutional neural networks can be thought of as methods that exploit the manifold structure of signals. While the empirical performance of these algorithms is impressive, the understanding of recovery of the signals and functions that live on manifold is less understood. In this paper, we focus on the recovery of signals that live on a union of surfaces. In particular, we consider signals living on a union of smooth band-limited surfaces in high dimensions. We show that an exponential mapping transforms the data to a union of low-dimensional subspaces. Using this relation, we introduce a sampling theoretical framework for the recovery of smooth surfaces from few samples and the learning of functions living on smooth surfaces. The low-rank property of the features is used to determine the number of measurements needed to recover the surface. Moreover, the low-rank property of the features also provides an efficient approach, which resembles a neural network, for the local representation of multidimensional functions on the surface.
