In this paper, we propose a spectral method for deriving functions that are jointly smooth on multiple observed manifolds. This allows us to register measurements of the same phenomenon by heterogeneous sensors, and to reject sensor-specific noise. Our method is unsupervised and primarily consists of two steps. First, using kernels, we obtain a subspace spanning smooth functions on each separate manifold. Then, we apply a spectral method to the obtained subspaces and discover functions that are jointly smooth on all manifolds. We show analytically that our method is guaranteed to provide a set of orthogonal functions that are as jointly smooth as possible, ordered by increasing Dirichlet energy from the smoothest to the least smooth. In addition, we show that the extracted functions can be efficiently extended to unseen data using the Nystr\"{o}m method. We demonstrate the proposed method on both simulated and real measured data and compare the results to nonlinear variants of the seminal Canonical Correlation Analysis (CCA). Particularly, we show superior results for sleep stage identification. In addition, we show how the proposed method can be leveraged for finding minimal realizations of parameter spaces of nonlinear dynamical systems.

翻译：在本文中, 我们提出一个光谱方法, 用来得出在多个已观测到的元体上共同平稳的函数。 这使我们能够通过不同传感器登记对同一现象的测量, 并拒绝传感器特有的噪音。 我们的方法不受监督, 主要由两步组成 。 首先, 使用内核, 我们获得一个子空间, 覆盖每个独立元体上的光滑功能 。 然后, 我们对获得的子空间应用光谱方法, 并发现所有元体上共同平稳的函数 。 我们从分析中显示, 我们的方法保证提供一套尽可能共同平稳的正方形函数, 由最顺畅的传感器来增加 Dirichlet 能量。 此外, 我们还显示, 提取的功能可以有效地扩展至看不见的数据, 使用 Nystr\ { o}m 方法 。 我们演示了模拟和真实测量的数据的拟议方法, 并将结果与半Canic Corrolational 分析( CACATI) 的非线性变量进行比较 。 特别是, 我们为睡眠阶段的识别显示优异差的结果 。 此外, 我们展示了如何利用拟议方法来找到最起码的动态空间参数的参数。