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 are also shown.

翻译：去噪、最小二乘期望以及流形学习等不同任务，通常可归结为从两个随机变量的乘积中求取条件期望这一共同框架。本文聚焦于这一更具普遍性的问题，并提出一种基于算子理论的条件期望估计方法。我们采用核积分算子作为压缩化工具，将估计问题转化为再生核希尔伯特空间中的线性逆问题。研究表明，该方程的解可实现数值逼近，从而保证数据驱动实现方法的收敛性。该技术易于实现，并展示了其在真实问题中的成功应用。