SIAM Journal on Imaging Sciences, Volume 14, Issue 2, Page 580-619, January 2021.
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 the 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. The direct representation of such a function in high dimensions often suffers from the curse of dimensionality; the large number of parameters would translate to the need for extensive training data. The low-rank property of the features can significantly reduce the number of parameters, which makes the computational structure attractive for learning and inference from limited labeled training data.

中文翻译：

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高维表面和函数的恢复：采样理论和神经网络的链接


SIAM 影像科学杂志，第 14 卷，第 2 期，第 580-619 页，2021 年 1 月。

多种成像算法，包括基于块的图像去噪、图像时间序列恢复和卷积神经网络，都可以被认为是利用信号流形结构的方法。虽然这些算法的经验性能令人印象深刻，但对流形上的信号和函数的恢复的理解却很少。在本文中，我们重点关注表面联合上信号的恢复。特别是，我们考虑存在于高维光滑带限表面的联合上的信号。我们证明指数映射将数据转换为低维子空间的并集。利用这种关系，我们引入了一个采样理论框架，用于从少量样本中恢复光滑表面以及学习光滑表面上的函数。特征的低阶属性用于确定恢复表面所需的测量次数。此外，特征的低秩属性还提供了一种类似于神经网络的有效方法，用于表面上多维函数的局部表示。在高维中直接表示此类函数常常会遭受维数灾难。大量的参数将意味着需要大量的训练数据。特征的低秩特性可以显着减少参数的数量，这使得计算结构对于从有限的标记训练数据中进行学习和推理具有吸引力。