The separate tasks of denoising, least squares expectation, and manifold learning can often be posed in a common setting of finding the conditional expectations arising from a product of two random variables. This paper focuses on this more general problem and describes an operator theoretic approach to estimating the conditional expectation. Kernel integral operators are used as a compactification tool, to set up the estimation problem as a linear inverse problem in a reproducing kernel Hilbert space. This equation is shown to have solutions that allow numerical approximation, thus guaranteeing the convergence of data-driven implementations. The overall technique is easy to implement, and their successful application to some real-world problems is also shown.

中文翻译：

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使用压缩运算符的条件期望

去噪、最小二乘期望和流形学习的单独任务通常可以在寻找由两个随机变量的乘积产生的条件期望的共同设置中提出。本文重点关注这个更普遍的问题，并描述了估计条件期望的算子理论方法。核积分算子用作压缩工具，将估计问题设置为再生核希尔伯特空间中的线性逆问题。该方程被证明具有允许数值近似的解，从而保证了数据驱动实现的收敛性。整体技术很容易实现，并且还展示了它们在一些实际问题上的成功应用。